Sindbad~EG File Manager
/* Initialize menu text for MathXpert */
/* Translator: translate text enclosed in quotation marks,
but do NOT translate text (usually formulas)
enclosed in dollar signs. Use the ISO-Latin1
character set.
*/
#define ENGLISH_DLL
#include "export.h" /* do not translate this or the next 3 lines */
#include "mtext.h"
#include "operator.h"
#include "english1.h"
static const char arithstr[] = "aritm�tica"; /* save space with ONE copy of this */
const char *Spanish_menutext[MAXMENUS][MAXLENGTH] =
{
{ /* numerical_calculation */
arithstr,
"C�lculo en n�meros decimales",
"Evaluaci�n decimal de $\\sqrt $ o de $^n\\sqrt $",
"C�lculo decimal de $x^n$",
"C�lculo decimal de valores de una funci�n",
"Factorizaci�n de n�meros enteros",
"Evaluaci�n num�rica en un punto",
"Aproximaci�n decimal $\\pi $",
"Aproximaci�n decimal del valor num�rico de e",
"C�lculo de valores de una funci�n",
"Factorizaci�n num�rica de un polinomio",
"Evaluar el n�mero de Bernoulli exactamente",
"Evaluar el n�mero de Euler exactamente"
},
{ /* numerical_calculation2 */
"Transformaci�n de decimal a fracci�n",
"Expresi�n como cuadrado",
"Expresi�n como cubo",
"Expresi�n como potencia e$n$-�sima",
"Expresi�n como potencia de ?",
"Notaci�n de un entero en el formato a^n",
"x = ? + (x-?)"
},
{ /* complex_arithmetic */
"$i^2 = -1$",
"i^(4n) = 1",
"i^(4n+1) = i",
"i^(4n+2) = -1",
"i^(4n+3) = -i",
"Aritm�tica compleja",
"Potencia de un n�mero complejo",
"Aritm�tica compleja y potencias",
"C�lculo con n�meros complejos decimales",
"Factorizaci�n en el conjunto de los enteros",
"Factorizaci�n de enteros por n�meros complejos",
"Factorizaci�n de n+mi (siendo n distinto de cero)",
"Aproximaci�n decimal de $\\sqrt $ o de $^n\\sqrt $",
"Valor num�rico decimal de $x^n$",
"Aproximaci�n decimal de valores de una funci�n",
"Evaluaci�n num�rica en un punto"
},
{ /* simplify_sums */
"Simplificar cada signo menos duplicado -(-a)=a",
"Distribuci�n de un signo menos en -(a+b) = -a-b",
"-a-b = -(a+b)",
arithstr,
"Organizaci�n de los t�rminos",
"Orden de los t�rminos",
"Eliminaci�n de t�rminos nulos, x+0 = x",
"Cancelaci�n de t�rminos anulados de a pares, $\\pm $",
"Agrupaci�n de t�rminos id�nticos con signo previo $\\pm $",
"Reagrupamiento de a dos, de sumandos id�nticos respecto del signo $\\pm $",
"a+b = b+a",
"a(b-c) = -a(c-b)",
"-ab = a(-b)",
"-abc = ab(-c)",
"a(-b)c = ab(-c)"
},
{ /*simplify_products */
"$x\\times 0 = 0\\times x = 0$",
"$x\\times 1 = 1\\times x = x$",
"a(-b) = -ab",
"a(-b-c) = -a(b+c)",
"(-a-b)c = -(a+b)c",
"Regrupamiento de factores",
"Reagrupamiento de n�meros",
"Ordenamiento de factores",
"Regrupamiento de potencias",
"a(b+c)=ab+ac",
"$(a-b)(a+b) = a^2-b^2$",
"$(a + b)^2 = a^2 + 2ab + b^2$",
"$(a - b)^2 = a^2 - 2ab + b^2$",
"$(a-b)(a^2+ab+b^2)=a^3-b^3$",
"$(a+b)(a^2-ab+b^2)=a^3+b^3$",
"ab = ba"
},
{ /* expand_menu */
"Desarrollo de productos de sumas",
"Multiplicaci�n del numerador",
"Multiplicaci�n del denominador",
"na = a +...+ a"
},
{ /* fractions */
"0/a = 0",
"a/1 = a",
"a(1/a) = 1",
"Multiplicaci�n de fracciones (a/c)(b/d)=ab/cd",
"a(b/c) = ab/c",
"Simplificaci�n ab/ac = b/c",
"Suma de fracciones $a/c \\pm b/c=(a\\pm b)/c$",
"Distribuci�n $(a \\pm b)/c = a/c \\pm b/c$",
"Distribuci�n y simplificaci�n $(ac\\pm b)/c = a\\pm b/c$",
"Divisi�n polinomial",
"Simplificaci�n aplicando una divisi�n polinomial",
"au/bv=(a/b)(u/v) (a,b enteros)",
"a/b = (1/b) a",
"au/b=(a/b)u (n�meros reales a,b)",
"ab/cd = (a/c)(b/d)",
"ab/c = (a/c) b"
},
{ /* signed_fractions */
"(-a)/(-b) = a/b",
"-(a/b) = (-a)/b",
"-(a/b) = a/(-b)",
"(-a)/b = -(a/b)",
"a/(-b)= -a/b",
"(-a-b)/c = -(a+b)/c",
"a/(-b-c) = -a/(b+c)",
"a/(b-c) = -a/(c-b)",
"-a/(-b-c) = a/(b+c)",
"-a/(b-c) = a/(c-b)",
"-(-a-b)/c = (a+b)/c",
"$$(a-b)/(c-d) = (b-a)/(d-c)$$",
"ab/c = a(b/c)",
"a/bc = (1/b) (a/c)"
},
{ /* compound_fractions */
"(a/c)/(b/c) = a/b",
"a/(b/c)=ac/b (inversi�n y multiplicaci�n)",
"1/(a/b) = b/a",
"(a/b)/c = a/(bc)",
"(a/b)/c = (a/b)(1/c)",
"(a/b)c/d = ac/bd",
"Factorizaci�n del denominador",
"Determinaci�n del com�n denominador de las fracciones",
},
{ /* common_denominators */
"Factorizaci�n del denominador",
"Determinaci�n del denominador com�n",
"Determinaci�n de un denominador com�n (solo fracciones)",
"Multiplicaci�n de fracciones (a/b)(c/d)=ac/bd",
"Multiplicaci�n de fracciones a(c/d)= ac/d",
"Ordenamiento de los factores",
"Suma de fracciones $a/c \\pm b/c=(a \\pm b)/c$",
"Determinaci�n del denominador com�n",
"Determinaci�n del denominador com�n (solo fracciones)",
"Determinaci�n del denominador com�n y simplificaci�n del numerador",
"Determinaci�n del denominador com�n y simplificaci�n (solo fracciones)",
"Multiplicaci�n del numerador y del denominador �por ?"
},
{ /* exponents */
"a^0 = 1 (a no nulo)",
"a^1 = a",
"0^b = 0 si b > 0",
"1^b = 1",
"$(-1)^n = \\pm 1$ (n par o impar)",
"(a^b)^c = a^(bc) si a>0 o $c\\in Z$",
"$(-a)^n = (-1)^na^n$",
"$(a/b)^n = a^n/b^n$",
"$(ab)^n = a^nb^n$",
"$(a+b)^2 = a^2+2ab+b^2$",
"Desarrollo aplicando la f�rmula del binomio",
"Reagrupamiento de potencias",
"a^(b+c) = a^b a^c", /* reversecollectpowers */
"$a^n/b^n = (a/b)^n$",
"b^n/b^m = b^(n-m)",
"ab^n/b^m = a/b^(m-n)"
},
{ /* expand_powers */
"a^2 = aa",
"a^3 = aaa",
"a^n = aaa...(n veces)",
"a^n = a^?a^(n-?)",
"$(a \\pm b)^2 = a^2 \\pm 2ab + b^2$",
"(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3",
"(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3",
"a^(bc) = (a^b)^c si $a>0$ o $c\\in Z$",
"a^(bc) = (a^c)^b si $a>0$ o $c\\in Z$",
"a^(b?) = (a^b)^?",
"1/a^n = (1/a)^n"
},
{ /* negative_exponents */
"a^(-n) = $1/a^n$ (n constante)",
"$a^(-n)/b = 1/(a^nb)$ (n constante)",
"a^(-1) = 1/a",
"$a^(-n) = 1/a^n$",
"$a^(-n)/b = 1/(a^nb)$",
"a/b^(-n) = ab^n",
"$a/b^n = ab^(-n)$",
"a/b = ab^(-1)",
"$(a/b)^(-n) = (b/a)^n$",
"b^n/b^m = b^(n-m)",
"ab^n/b^m = a/b^(m-n)",
"a^(b-c) = a^b/a^c"
},
{ /* square_roots */
"$\\sqrt x\\sqrt y = \\sqrt (xy)$",
"$\\sqrt (xy) = \\sqrt x\\sqrt y$",
"$\\sqrt (x^2y) = x\\sqrt y$ o $|x|\\sqrt y$",
"$\\sqrt (x^2)=x$ si $x\\ge 0$",
"$\\sqrt (x^2)=|x|$",
"Factorizaci�n del entero x en $\\sqrt x$",
"$\\sqrt (x/y) = \\sqrt x/\\sqrt y$",
"$\\sqrt (x/y) = \\sqrt |x|/\\sqrt |y|$",
"$\\sqrt x/\\sqrt y = \\sqrt (x/y)$",
"$x/\\sqrt x = \\sqrt x$",
"$\\sqrt x/x = 1/\\sqrt x$",
"$(\\sqrt x)^2^n = x^n$ si $x\\ge 0$",
"$(\\sqrt x)^(2n+1) = x^n\\sqrt x$",
"C�lculo por aproximaci�n racional de $\\sqrt $",
"C�lculo por aproximaci�n decimal de $\\sqrt $",
"C�lculo por aritm�tica simple" /* that is, doesn't compute roots */
},
{ /* advanced_square_roots */
"Extracci�n del factor com�n en $\\sqrt u/\\sqrt v$",
"Factorizaci�n del polinomio bajo el signo $\\sqrt $",
"Racionalizaci�n del denominador",
"Racionalizaci�n del numerador",
"$\\sqrt (x^2)=|x|$ o $\\sqrt (x^2^n)=|x|^n$",
"Simplificaci�n de $\\sqrt $: $\\sqrt (xy)/\\sqrt y = \\sqrt x$",
"Multiplicaci�n bajo el signo de $\\sqrt $",
"$a^2-b = (a-\\sqrt b)(a+\\sqrt b)$",
"$^2\\sqrt u = \\sqrt u$",
"$\\sqrt u = ^2^n\\sqrt u^n$",
"$\\sqrt u = (^2^n\\sqrt u)^n$",
"$\\sqrt (u^2^n) = u^n$ si $u^n\\ge 0$",
"$\\sqrt (u^(2n+1)) = u^n\\sqrt u$ si $u^n\\ge 0$",
"$a\\sqrt b = \\sqrt (a^2b)$ si $a\\ge 0$",
"Racionalizaci�n del denominador y simplificaci�n"
},
{ /* fractional_exponents */
"$a ^ \\onehalf = \\sqrt a$",
"$a^(n/2) = \\sqrt (a^n)$",
"$a^(b/n) = ^n\\sqrt (a^b)$",
"$\\sqrt a = a ^ \\onehalf $",
"$^n\\sqrt a = a^(1/n)$",
"$^n\\sqrt (a^m) = a^(m/n)$",
"$(^n\\sqrt a)^m = a^(m/n)$",
"$(\\sqrt a)^m = a^(m/2)$",
"$1/\\sqrt a = a^(-\\onehalf )$",
"$1/^n\\sqrt a = a^(-1/n)$",
"Evaluaci�n de (-1)^(p/q)",
"Factorizaci�n entera de a en a^(p/q)",
"a/b^(p/q) = (a^q/b^p)^(1/q)",
"a^(p/q)/b = (a^p/b^q)^1/q)",
"$a^(n/2) = (\\sqrt a)^n$",
"$a^(m/n) = (^n\\root a)^m$"
},
{ /*nth_roots */
"$^n\\sqrt x^n\\sqrt y = ^n\\sqrt (xy)$",
"$^n\\sqrt (xy) = ^n\\sqrt x ^n\\sqrt y$",
"$^n\\sqrt x^m = (^n\\sqrt x)^m$ si $x\\ge 0$ o n impar", /* rootofpower5 */
"$^n\\sqrt (x^ny) = x ^n\\sqrt y$ o $|x|^n\\sqrt y$",
"$^n\\sqrt (x^n) = x$ si $x\\ge 0$ o n impar", /* rootofpower */
"$^n\\sqrt (x^(nm))=x^m$ si $x\\ge 0$ o n impar", /* rootofpower3 */
"$^2^n\\sqrt (x^n) = \\sqrt x$", /* rootofpower2 */
"$^m^n\\sqrt x^m) = ^n\\sqrt x$", /* rootofpower4 */
"$(^n\\sqrt x)^n = x$", /* powerofroot */
"$(^n\\sqrt a)^m = ^n\\sqrt (a^m)$", /* powerofroot2 */
"$(^n\\sqrt a)^(qn+r) = a^q ^n\\sqrt (a^r)$", /* powerofroot3 */
"Factorizaci�n del x entero en $^n\\sqrt x$", /* factorunderroot */
"$^n\\sqrt (-a) = -^n\\sqrt a$, n impar",
"C�lculo por aproximaci�n racional",
"Factorizaci�n del polinomio bajo el signo $^n\\sqrt $",
"Multiplicaci�n bajo el signo $^n\\sqrt $"
},
{ /* roots_of_roots */
"$\\sqrt (\\sqrt x) = ^4\\sqrt x$", /* sqrtofsqrt */
"$\\sqrt (^n\\sqrt x) = ^2^n\\sqrt x$", /* sqrtofroot */
"$^n\\sqrt (\\sqrt x) = ^2^n\\sqrt x$", /* rootofsqrt */
"$^n\\sqrt (^m\\sqrt x) = ^n^m\\sqrt x$", /* rootofsqrt */
},
{ /* roots_and_fractions */
"$^n\\sqrt (x/y) = ^n\\sqrt x/^n\\sqrt y$",
"$^n\\sqrt x/^n\\sqrt y = ^n\\sqrt (x/y)$",
"$x/^n\\sqrt x = (^n\\sqrt x)^(n-1)$",
"$^n\\sqrt x/x = 1/(^n\\sqrt x)^(n-1)$",
"Simplificaci�n respecto del signo $^n\\sqrt: ^n\\sqrt (ab)/^n\\sqrt (bc)=^n\\sqrt a/^n\\sqrt b$",
"Simplificaci�n respecto del signo $^n\\sqrt $: $^n\\sqrt (xy)/^n\\sqrt y = ^n\\sqrt x$",
"Extracci�n del factor com�n en $^n\\sqrt u/^n\\sqrt v$",
"$a(^n\\sqrt b) = ^n\\sqrt (a^nb)$ si n impar",
"$a(^n\\sqrt b) = ^n\\sqrt (a^nb)$ si $a\\ge 0$",
"$-^n\\sqrt a = ^n\\sqrt (-a)$ si n impar",
"$a/^n\\sqrt b = ^n\\sqrt (a^n/b)$ (n impar o $a\\ge 0$)",
"$^n\\sqrt a/b = ^n\\sqrt (a/b^n)$ (n impar o $b>0$)",
"$\\sqrt a/b = \\sqrt (a/b^2)$ si $b>0$",
"$a/\\sqrt b = \\sqrt (a^2/b)$ si $a\\ge 0$",
"$(^m^n\\sqrt a)^n = ^m\\sqrt a$",
"$(^2^n\\sqrt a)^n = \\sqrt a$"
},
{ /* complex_numbers */
"1/i = -i",
"a/i = -ai",
"a/(bi) = -ai/b",
"$\\sqrt (-1) = i$",
"$\\sqrt (-a) = i\\sqrt a$ si $a\\ge 0$",
"Extracci�n de la parte imaginaria i del denominador",
"$(a-bi)(a+bi) = a^2+b^2$",
"$a^2+b^2 = (a-bi)(a+bi)$",
"$|u + vi|^2 = u^2 + v^2$",
"$|u + vi| = \\sqrt (u^2+v^2)$",
"(u+vi)/w = u/w + (v/w)i",
"Notaci�n en formato u+vi",
"$\\sqrt(bi)= \\sqrt(b/2)+\\sqrt(b/2)i$, si b >= 0",
"$\\sqrt(-bi)= \\sqrt(b/2)-\\sqrt(b/2)i$, si b >= 0",
"$\\sqrt(a+bi)= \\sqrt((a+c)/2)+\\sqrt((a-c)/2)i$, si b \\ge 0 y $c^2=a^2+b^2$",
"$\\sqrt(a-bi)= \\sqrt((a+c)/2)-\\sqrt((a-c)/2)i$, si b \\ge 0 y $c^2=a^2+b^2$"
},
{ /* factoring */
"Factorizaci�n del n�mero",
"Extracci�n de denominadores num�ricos",
"ab + ac = a(b+c)",
"Factorizaci�n de la potencia mayor",
"$a^2+2ab+b^2 = (a+b)^2$",
"$a^2-2ab+b^2 = (a-b)^2$",
"$a^2-b^2 = (a-b)(a+b)$",
"Factorizaci�n del trinomio de segundo grado",
"Aplicaci�n de la f�rmula de resoluci�n de las ecuaciones de segundo grado",
"$a^2^n = (a^n)^2$",
"$a^nb^n = (ab)^n$",
"Factorizaci�n del coeficientes enteros",
"Factorizaci�n de un entero",
"Sustituci�n por cambio de una variable, u = ?",
"Eliminaci�n de una variable que ya se ha definido",
"Consideraci�n de una variable como constante",
},
{ /* advanced_factoring */
"Formulaci�n como funci�n �de ?",
"Formulaci�n como funci�n �de ? y �de ?",
"a^(3n) = (a^n)^3",
"a^(?n) = (a^n)^?",
"a^3 - b^3 = (a-b)(a^2+ab+b^2)",
"a^3 + b^3 = (a+b)(a^2-ab+b^2)",
"$a^n-b^n = (a-b)(a^(n-1)+...+b^(n-1))$",
"$a^n-b^n = (a+b)(a^(n-1)-...-b^(n-1))$ (n par)",
"$a^n+b^n=(a+b)(a^(n-1)-...+b^(n-1))$ (n impar)",
"$x^4+a^4=(x^2-\\sqrt 2ax+a^2)(x^2+\\sqrt 2ax+a^2)$",
"$x^4+(2p-q^2)x^2+p^2=(x^2-qx+p)(x^2+qx+p)$",
"Sustituci�n por cambio de variable a cargo de MathXpert",
"Ensayo de un factor",
"B�squeda de un factor lineal",
"Factorizaci�n por aprupamiento",
"Formulaci�n como polinomio �en ?",
},
{ /* solve_equations */
"Cambio de miembros",
"Cambio de signo en ambos miembros",
"Suma �de ? en ambos miembros miembros",
"Resta �de ? en ambos miembros miembros",
"Pasaje �de ? izquierda a derecha",
"Pasaje �de ? derecha a izquierda",
"Multiplicaci�n de ambos miembros �por ?",
"Divisi�n de ambos miembros �por ?",
"Potencia cuadrada de ambos miembros",
"Simplificaci�n de t�rminos $\\pm $ id�nticos de ambos miembros",
"Simplificaci�n del factor com�n de ambos miembros",
"Sustracci�n para obtener una ecuaci�n de la forma u=0",
"Ecuaci�n id�ntica de valor de verdad cieto",
"a=-b se convierte en $a^2=-b^2$ si $a,b\\ge 0$",
"a=-b se convierte en a=0 si $a,b\\ge 0$",
"a=-b se convierte en b=0 si $a,b\\ge 0$"
},
{ /* quadratic_equations */
"Si ab=0 entonces a=0 o b=0",
"Aplicaci�n de la f�rmula de resoluci�n de las ecuaciones de segundo grado",
"$x = -b/2a \\pm \\sqrt (b^2-4ac)/2a$",
"Operaci�n para completar el cuadrado",
"Extracci�n de la ra�z cuadrada de ambos miembros",
"Multiplicaci�n cruzada",
"Si $b^2-4ac < 0$ no hay ra�ces reales",
"[p=a,p=-a] se convierte en p=|a| (dado que $p\\ge 0$)",
arithstr
},
{ /* numerical_equations */
"Evaluaci�n num�rica calculada en un punto",
"Resoluci�n num�rica"
},
{ /* advanced_equations */
"Multiplicaci�n cruzada (a/b=c/d => ad=bc)",
"Si u=v entonces $u^n=v^n$",
"Aplicaci�n en ambos miembros de la funci�n $\\sqrt $",
"Aplicaci�n en ambos miembros de la funci�n $^n\\sqrt $",
"Aplicaci�n en ambos miembros �de la funci�n ?",
"Denominador com�n",
"Si ab=0 entonces a=0 o b=0",
"Si ab=ac entonces a=0 o b=c",
"Visualizaci�n restringida a la ecuaci�n seleccionada",
"Visualizaci�n del conjunto de todas las ecuaciones, recuperada",
"Reagrupamiento de las soluciones m�ltiples",
"Sustituci�n por cambio de �variable u = ?",
"Eliminaci�n de una variable que ya se ha definido",
"Pechazo de una ecuaci�n irresoluble",
"Verificaci�n de las ra�ces en la ecuaci�n de partida",
"Resoluci�n inmediata de una ecuaci�n lineal, en un solo paso",
},
{ /* cubic_equations */
"u=x+b/3 en ax^3+bx^2+cx+d=0",
"C�lculo del discriminante", /* if this changes change discriminant_line in cubics.c */
"Visualizaci�n de la ecuaci�n c�bica, recuperada",
"Cambio de la variable de Vieta x=y-a/3cy en cx^3+ax+b=0",
"F�rmula de Cardan, 1 ra�z real",
"F�rmula de Cardan, 3 ra�ces reales",
"F�rmula de Cardan, ra�ces complejas",
"Sustituci�n de x = f(u)",
"Eliminaci�n de una variable definida",
"Sustituci�n de n = ?-k",
"Determinaci�n exacta de las ra�ces reales",
"C�lculo del conjunto de n�meros decimales",
"Simplificaci�n",
},
{ /* logarithmic_equations */
"Si u=v entonces a^u = a^v",
"Si ln u = v entonces u = e^v",
"Si log u = v entonces u = 10^v",
"Si log(b,u) = v entonces u = b^v",
"Si a^u = a^v entonces u=v",
"Extracci�n del log de ambos miembros",
"Extracci�n del logaritmo natural de ambos miembros",
"Rechazo de la ecuaci�n imposible de log o ln",
},
{ /* cramers_rule */
"Regla de Cramer",
"C�lculo del determinante"
},
{ /* several_linear_equations*/
"Ordenar variables a derecha y constantes a izquierda",
"Agrupaci�n de t�rminos similares", /* if position changes, change exec.c (search for "several_linear_equations") */
"Organizaci�n alineada de las variables",
"Suma de dos ecuaciones",
"Resta de dos ecuaciones",
"Multiplicaci�n de la ecuaci�n ? por ?",
"Division de la ecuaci�n ? por ?",
"Suma de un m�ltiplo de la ecuaci�n ? a la ecuaci�n ?",
"Resta de un m�ltiplo de la ecuaci�n ? de la ecuaci�n ?",
"Intercambio de dos ecuaciones",
"Reordenamiento de las ecuaciones ya resueltas",
"Eliminaci�n de las identidades",
"Consideraci�n de una variable como una constante",
"Contradicci�n: no hay soluci�n",
},
{ /* selection_mode_only */
"a|b| = |ab| si $0 \\le a$",
"|b|/c = |b/c| si 0 < c",
"a|b|/c = |ab/c| si 0 <a/c",
"Resoluci�n para ?" /* solvelinearfor */
},
{ /* linear_equations_by_selection */
"Suma de la ecuaci�n seleccionada a la ecuaci�n ?",
"Resta de la ecuaci�n seleccionada a la ecuaci�n ?",
"Multiplicaci�n de la ecuaci�n seleccionada por la ecuaci�n ?",
"Dividisi�n de la ecuaci�n seleccionada por ?",
"Suma de un m�ltiplo de la ecuaci�n seleccionada a la ecuaci�n ?",
"Resta de un m�ltiplo de la ecuaci�n seleccionada a la ecuaci�n ?",
"Intercambio de la ecuaci�n seleccionada por la ecuaci�n ?",
"Resoluci�n de la ecuaci�n seleccionada respecto de ?",
"Suma de la fila seleccionada a la fila ?",
"Resta de la fila seleccionada a la fila ?",
"Multiplicaci�n de la fila seleccionada por ?",
"Dividisi�n de la fila seleccionada por ?",
"Suma de un m�ltiplo de la fila seleccionada a la fila ?",
"Resta de un m�ltiplo de la fila seleccionada a la fila ?",
"Intercambio de la fila seleccionada por la fila ?",
"A = IA"
},
{ /* linear_equations_by_substitution */
"Agrupaci�n de t�rminos similares",
"Resoluci�n de la ecuaci�n ? respecto de ?",
"Simplificaci�n de ecuaciones",
"Simplificaci�n de los t�rminos presentes en ambos miembros",
"Suma de ? en ambos miembros de la ecuaci�n ?",
"Resta de ? de ambos miembros de la ecuaci�n ?",
"Divisi�n de la ecuaci�n ? por ?",
"Sustituci�n de la variable de una funci�n",
"Contradicci�n evidente: no hay soluci�n",
},
{ /* matrix_methods */
"Expresi�n en forma matricial",
"A = IA",
"Intercambio de filas",
"Suma de filas",
"Resta de una fila a la otra",
"Multiplicaci�n de una fila por una constante",
"Divisi�n de una fila por una constante",
"Suma de un m�ltiplo de una fila a otra",
"Suma de un m�ltiplo de una fila de otra",
"Multiplicaci�n de matrices",
"Eliminaci�n de una columna nula",
"Eliminaci�n de una fila nula",
"Eliminaci�n de una fila duplicada",
"Contradicci�n evidente: no hay soluci�n",
"Conversi�n de un sistema de ecuaciones",
},
{ /* advanced_matrix_methods */
"Multiplicaci�n de matrices",
"AX = B se convierte en X = A^(-1)B",
"Aplicaci�n de la f�rmula de inversi�n de matrices de 2x2",
"C�lculo exacto de la matriz inversa",
"C�lculo decimal aproximado de la matriz inversa",
},
{ /* absolute_value */
"|u| = u si $u\\ge 0$",
"La hip�tesis seg�n la cual $u\\ge 0$, permite expresar que |u| = u",
"|u| = -u si $u\\le 0$",
"|cu| = c|u| si $c\\ge 0$",
"|u/c| = |u|/c si c>0",
"|u||v| = |uv|",
"|uv| = |u||v|",
"|u/v| = |u| / |v|",
"|u| / |v| = |u/v|",
"$|u|^2^n=u^2^n$ si u es real",
"$|u^n|=|u|^n$ si n es real",
"$|\\sqrt u| = \\sqrt |u|$",
"$|^n\\sqrt u| = ^n\\sqrt |u|$",
"|ab|/|ac| = |b|/|c|",
"|ab|/|a| = |b|",
"Extracci�n del factor com�n en |u|/|v|"
},
{ /* absolute_value_ineq1 */
"Si $c > 0$, entonces |u|=c si y solo si u=c o u = -c", /* abseqn */
"|u|/u = c si y solo si c = $\\pm $1", /* abseqn2 */
"|u| < v si y solo si -v < u < v", /* abslessthan */
"$|u| \\le v$ si y solo si $-v \\le u \\le v$", /* absle */
"u < |v| si y solo si v < -u o u < v", /* lessthanabs */
"$u \\le |v|$ si y solo si $v \\le -u$ o $u \\le v$",/* leabs */
"|u| = u si y solo si $0 \\le u$", /* abseqntoineq1 */
"|u| = -u si y solo si si $u \\le 0$", /* abseqntoineq2 */
"$0 \\le |u|$ es siempre cierto", /* absineqtrue */
"|u| < 0 es siempre falso", /* absineqfalse */
"Si $c\\le 0$, entonces $-c \\le |u|$", /* absineqtrue2 */
"Si c>0, entonces -c < |u|", /* absineqtrue3 */
"Si $c\\ge 0$, entonces |u| < -c es falso", /* abslessthanneg*/
"Si c>0, entonces $|u| \\le -c$ es falso", /* absleneg */
"Si $c\\ge 0$, entonces $|u| \\le -c$ si y solo si u=0", /* absleneg2 */
"Si $c\\ge 0$, entonces |u| = -c si y solo si u=0" /* abseqnneg */
},
{ /* absolute_value_ineq2 */
"v > |u| si y solo si -v < u < v", /* absgreaterthan */
"$v \\ge |u|$ si y solo si $-v \\le u \\le v$", /* absge */
"|v| > u si y solo si v < -u o v > u", /* greaterthanabs */
"$|v| \\ge u$ si y solo si $v \\le -u$ o $v \\ge u$",/* geabs */
"$|u| \\ge 0$ es cierto", /* absineqtrueg */
"0 > |u| es falso", /* absineqfalseg */
"-c > |u| es falso si $c\\ge 0$", /* absgreaterthanneg */
"$-c ? |u|$ es falso si $c>0$", /* absgeneg */
"Si $c\\ge 0$, entonces $-c \\ge |u|$ si y solo si u=0", /* absgeneg2 */
"Si c>0, entonces |u| > -c es cierto", /* absineqtrue3g */
"Si ($c\\ge 0$), entonces $|u| \\ge -c$ es cierto", /* absineqtrue2g */
"$-v \\le u \\le v$ si y solo si $|u| \\le v$ ", /* intervalabs1 */
"v < -u o u < v si y solo si u < |v|", /* intervalabs2 */
"Para todo real u, $u^(2n) = |u|^(2n)$", /* absevenpowerrev */
"Para todo real u, $u|^n = |u^n|$ si n es real" /* abspowerrev */
},
{ /* less_than */
"Cambio de u < v en v > u",
"Suma de ? en ambos miembros",
"Resta de ? de ambos miembros",
"Cambio de -u < -v en v < u",
"Cambio de -u < -v en u > v",
"Multiplicaci�n de ambos miembros por ?",
"Multiplicaci�n de ambos miembros ?^2",
"Divisi�n de ambos miembros por ?",
"Evaluaci�n num�rica de la inecuaci�n",
"$a < x^(2n)$ si $a < 0$",
"Si $a \\le 0$, entonces $x^(2n) < a$ es falso",
"Potencia elevando al cuadrado ambos miembros no-negativos",
"Potencia elevando al cuadrado si un miembro es $\\ge $ 0",
"u < v o u = v si y solo si $u \\le v$",
"Combinaci�n de intervalos",
"Aplicaci�n de hip�tesis",
},
{ /* greater_than */
"Cambio de x > y en y < x",
"Cambio de -u > -v en u < v",
"Cambio de -u > -v en v > u",
"$x^2^n > a$ es cierto si $a < 0$",
"$a > x^2^n$ es falso si $a \\le 0$",
"Potencia elevando al cuadrado si un miembro es $\\ge $ 0",
"u > v o u = v si y solo si $u \\ge v$",
},
{ /* less_than_or_equals */
"Cambio de $x \\le y$ en $y \\ge x$",
"Suma de ? en ambos miembros",
"Resta de ? de ambos miembros",
"Cambio de $-u \\le -v$ en $v \\le u$",
"Cambio de $-u \\le -v$ en $u \\ge v$",
"Multiplicaci�n de ambos miembros por ?",
"Multiplicaci�n de ambos miembros por ?^2",
"Divisi�n de ambos miembros por ?",
"Evaluaci�n num�rica de la inecuaci�n",
"$a \\le x^2^n$ es cierto si $a \\le 0$",
"$x^2^n \\le a$ es falso si $a < 0$",
"Potencia elevando al cuadrado ambos miembros",
"Si $0 \\le v$, entonces $u \\le v$ si y solo si $u^2 \\le v^2$ o $u \\le 0$",
"Combinaci�n de intervalos",
"Aplicaci�n de hip�tesis"
},
{ /* greater_than_or_equals */
"Cambio de $x \\ge y$ en $y \\le x$",
"Cambio de $-u \\ge -v$ en $u \\le v$",
"Cambio de $-u \\ge -v$ en $v \\ge u$",
"$x^(2n) \\ge a$ es cierto si $a \\le 0$",
"$a \\ge x^(2n)$ es falso si $a < 0$",
"Si $0 \\le v$, entonces $v \\ge u$ si y solo si $v^2 \\ge u^2$ o $u \\le 0$"
},
{ /* square_ineq1 */
"$u^2 < a$ si y solo si $|u| < \\sqrt a$",
"$u^2 < a$ si y solo si $-\\sqrt a < u < \\sqrt a$",
"Si $0\\le a$, entonces $a < v^2$ si y solo si $?a < |v|$",
"$a < u^2$ si y solo si $u < -\\sqrt a$ o $\\sqrt a < u$",
"$a < u^2 < b$ si y solo si $-\\sqrt b<u<-\\sqrt a$ o $\\sqrt a<u<\\sqrt b$",
"Si $0<a$, entonces $-a < u^2 < b$ si y solo si $u^2 < b$",
"Si $0<a$, entonces $-a < u^2 \\le b$ si y solo si $u^2 \\le b$",
"$\\sqrt u < v$ si y solo si $0 \\le u < v^2$",
"$0 \\le a\\sqrt u < v$ si y solo si $0 \\le a^2u < v^2$",
"Si $0\\le a$, $a < \\sqrt v$ si y solo si $a^2 < v$",
"$0 \\le u < v$ si y solo si $\\sqrt u < \\sqrt v$",
"Si a < 0, $a < x^2$ es cierto",
"Si $a \\le 0$, $x^2 < a$ es falso",
"Si a < 0, entonces $a < \\sqrt u$ si y solo si $0 \\le u$"
},
{ /* square_ineq2 */
"$u^2 \\le a$ si y solo si $|u| \\le \\sqrt a$",
"$u^2 \\le a$ si y solo si $-\\sqrt a \\le u \\le \\sqrt a$",
"Si $0\\le a$, entonces $a \\le v^2$ si y solo si $\\sqrt a \\le |v|$",
"$a \\le u^2$ si y solo si $u \\le -\\sqrt a$ o $\\sqrt a \\le u$",
"$a \\le u^2 \\le b$ si y solo si $-\\sqrt b\\le u\\le -\\sqrt a$ o $\\sqrt a\\le u\\le \\sqrt b$",
"Si $0\\le a$, entonces $-a \\le u^2 \\le b$ si y solo si $u^2 \\le b$",
"Si $0\\le a$, entonces $-a \\le u^2 < b$ si y solo si $u^2 < b$",
"$\\le \\sqrt u \\le v$ si y solo si $0 \\le u \\le v^2$",
"$0 \\le a\\le \\sqrt u \\le v$ si y solo si $0 \\le a^2u \\le v^2$",
"Si $0\\le a$, entonces $a \\le \\sqrt v$ si y solo si $a^2 \\le v$",
"$0 \\le u \\le v$ si y solo si $\\sqrt u \\le \\sqrt v$",
"$x^2 > a$ es cierto si a < 0",
"$a > x^2$ es falso si $a \\le 0$",
"Si $a \\le 0$, entonces $a \\le \\sqrt u$ si y solo si $0 \\le u$"
},
{ /* recip_ineq1 */
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Siendo a,b > 0, a < 1/x < b si y solo si 1/b < x < 1/a,",
"Siendo a,b > 0, $a < 1/x \\le b$ si y solo si $1/b \\le x < 1/a$,",
"Siendo a,b > 0, -a < 1/x < -b si y solo si -1/b < x < -1/a,",
"Siendo a,b > 0, $-a < 1/x \\le -b$ si y solo si $-1/b \\le x < -1/a$",
"Siendo a,b > 0, -a < 1/x < b si y solo si x < - 1/a o 1/b < x",
"Siendo a,b > 0, $-a < 1/x \\le b$ si y solo si x < -1/a o $1/b \\le x$",
},
{ /* recip_ineq2 */
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Siendo a,b > 0, $a \\le 1/x < b$ si y solo si $1/b < x \\le 1/a$,",
"Siendo a,b > 0, $a \\le 1/x \\le b$ si y solo si $1/b \\le x < 1/a$",
"Siendo a,b > 0, $-a \\le 1/x < -b$ si y solo si $-1/b < x \\le -1/a$",
"Siendo a,b > 0, $-a \\le 1/x \\le -b$ si y solo si $-1/b \\le x \\le -1/a$",
"Siendo a,b > 0, $-a \\le 1/x < b$ si y solo si $x \\le - 1/a$ o 1/b < x",
"Siendo a,b > 0, $-a \\le 1/x \\le b$ si y solo si $x \\le -1/a$ o $1/b \\le x$"
},
{ /* root_ineq1 */
"u < v si y solo si $^n\\sqrt u < ^n\\sqrt v$ (n impar)",
"$u^(2n) < a$ si y solo si $|u| < ^(2n)a$",
"$u^(2n) < a$ si y solo si $-^(2n)\\sqrt a < u < ^(2n)\\sqrt a$",
"$0 \\le a < u^(2n)$ si y solo si $^(2n)\\sqrt a < |u|$",
"$a < u^2^n$ si y solo si $u < -^2^n\\sqrt a$ o $^2^n\\sqrt a < u$",
"$a<u^2^n<b$ si y solo si $-^2^n\\sqrt b<u<-^2^n\\sqrt a$ o $^2^n\\sqrt a<u<^2^n\\sqrt b$",
"$^2^n\\sqrt u < v$ si y solo si $0 \\le u < v^2^n$",
"$^n\\sqrt u < v$ si y solo si $u < v^n$ (n impar o $u\\ge 0$)",
"Si $0 \\le a(^n\\sqrt u)$, entonces $a(^n\\sqrt u) < v$ si y solo si $a^nu < v^n$",
"Si $0 \\le u$, entonces $u < ^n\\sqrt v$ si y solo si $u^n < v$",
"$u < v$ si y solo si $u^n < v^n$ (n impar >0)",
"u < v si y solo si $u^n < v^n$ (n > 0 y $0 \\le u$)",
"Si a < 0, entonces $a < ^(2n)\\sqrt u$ si y solo si $0 \\le u$",
},
{ /* root_ineq2 */
"$u \\le v$ $^n\\sqrt u \\le ^n\\sqrt v$ (n impar)",
"$u^2^n \\le a$ si y solo si $|u| \\le ^2^n\\sqrt a$",
"$u^2^n \\le a$ si y solo si $-^2^n\\sqrt a \\le u \\le ^2^n\\sqrt a$",
"$0 \\le a \\le u^2^n$ si y solo si $^2^n\\sqrt a \\le |u|$",
"$a \\le u^2^n$ si y solo si $u \\le -^2^n\\sqrt a$ o $^2^n\\sqrt a \\le u$",
"$a\\le u^2^n\\le b$ si y solo si $-^2^n\\sqrt b\\le u\\le -^2^n\\sqrt a$ o $^2^n\\sqrt a\\le u\\le ^2^n\\sqrt b$",
"$^2^n\\sqrt u \\le v$ si y solo si $0 \\le u \\le v^2^n$",
"$^n\\sqrt u \\le v$ si y solo si $u \\le v^n$ (n impar o $u\\ge 0$)",
"Si $0 \\le a(^n\\sqrt u)$, entonces $a(^n\\sqrt u) \\le v$ si y solo si $a^nu \\le v^n$",
"Si $0 \\le u$, entonces $u \\le ^n\\sqrt v$ si y solo si $u^n \\le v$",
"$u \\le v$ si y solo si $u^n \\le v^n$ (n impar, $n \\ge 0$)",
"$u \\le v$ si y solo si $u^n \\le v^n$ (n > 0 y $0 \\le u$)",
"Si $a \\le 0$, entonces $a \\le ^2^n\\sqrt u$ si y solo si $0 \\le u$"
},
{ /* zero_ineq1 */
"Eliminaci�n de factores estrictamente positivos",
"Si $u \\ge 0$, entonces $0 \\le u/v$ si y solo si $0 \\le v$",
"Cambio de $0 < u/\\sqrt v$ por 0 < uv",
"0 < u/v si y solo si 0 < uv",
"Cambio de $u/\\sqrt v < 0$ por uv < 0",
"u/v < 0 si y solo si uv < 0",
"$ax \\pm b < 0$ si y solo si $a(x\\pm b/a) < 0$",
"Cambio de u < v to v > u",
"Si a<b, entonces (x-a)(x-b) < 0 si y solo si a<x<b",
"Si a<b, entonces 0 < (x-a)(x-b) si y solo si x<a o b<x"
},
{ /* zero_ineq2 */
"Eliminaci�n de factores estrictamente positivos",
"Si $u \\ge 0$, entonces $0 \\le u/v$ si y solo si $0 \\le v$",
"$0 \\le u/\\sqrt v$ si y solo si $0 \\le uv$",
"$0 \\le u/v$ si y solo si 0 < uv o u = 0",
"$u/\\sqrt v \\le 0$ si y solo si $uv \\le 0$",
"$u/v \\le 0$ si y solo si uv < 0 o u = 0",
"$ax \\pm b \\le 0$ si y solo si $a(x\\pm b/a) \\le 0$",
"Cambio de $u \\le v$ por $v \\ge u$",
"Si $a\\le b$, entonces $(x-a)(x-b) \\le 0$ si y solo si $a\\le x\\le b$",
"Si $a\\le b$, entonces $0\\le (x-a)(x-b)$ si y solo si $x\\le a$ o $b\\le x$"
},
{ /* square_ineq3 */
"$a > u^2$ si y solo si $\\sqrt a > |u|$",
"$a > u^2$ si y solo si $-\\sqrt a < u < \\sqrt a$",
"Si $a\\ge 0$, entonces $v^2 > a$ si y solo si $|v| > \\sqrt a$",
"$u^2 > a$ si y solo si $u < -\\sqrt a$ o $u > \\sqrt a$",
"$v > \\sqrt u$ si y solo si $0 \\le u < v^2$",
"Si $0\\le a$, entonces $v>a\\sqrt u$ si y solo si $0\\le a^2u<v^2$",
"Si $0\\le a$, entonces $\\sqrt v > a$ si y solo si $v > a^2$",
"Si $u\\ge 0$, entonces v > u si y solo si $\\sqrt v > \\sqrt u$",
"$x^2 > a$ es cierto si $a < 0$",
"$a > x^2$ es falso si $a <= 0$",
"Si $a < 0$, entonces $\\sqrt u > a$ si y solo si $u \\ge 0$"
},
{ /* square_ineq4 */
"$a \\ge u^2$ si y solo si $6\\sqrt a \\ge |u|$",
"$a \\ge u^2$ si y solo si $-\\sqrt a \\le u \\le \\sqrt a$",
"Si $0\\le a$, entonces $v^2 \\ge a$ si y solo si $|v| \\ge \\sqrt a$",
"$u^2 \\ge a$ si y solo si $u \\le -\\sqrt a$ o $\\sqrt a \\le u$",
"$v \\ge \\sqrt u$ si y solo si $60 \\le u \\le v^2$",
"Si $0\\le a$, entonces $v \\ge a\\sqrt u$ si y solo si $0\\le a^2u\\le v^2$",
"Si $0\\le a$, entonces $\\sqrt v \\ge a$ si y solo si $v \\ge a^2$",
"Si $u\\ge 0$, entonces $v \\ge u$ si y solo si $\\sqrt v \\ge \\sqrt u$",
"$x^2 \\ge a$ es cierto si $a \\le 0$",
"$a \\ge x^2$ es falso si a < 0",
"Si $a\\le 0$, entonces $\\sqrt u \\ge a$ si y solo si $u \\ge 0$"
},
{ /* recip_ineq3 */
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
},
{ /* recip_ineq4 */
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
"Componer ambos miembros seg�n la funci�n inversa $(x -> 1/x)$",
},
{ /* root_ineq3 */
"$u > v$ si y solo si $^n\\sqrt u > ^n\\sqrt v$ (n impar)",
"$a > u^2^n$ si y solo si $^2^n\\sqrt a > |u|$",
"$a > u^2^n$ si y solo si $-^2^n\\sqrt a < u < ^2^n\\sqrt a$",
"Si $a\\ge 0$, entonces $u^2^n > a$ si y solo si $|u| > ^2^n\\sqrt a$",
"$u^2^n > a$ si y solo si $u < -^2^n\\sqrt a$ o $u > ^2^n\\sqrt a$",
"$v > ^2^n\\sqrt u$ si y solo si $0 \\le u < v^2^n$",
"$v > ^n\\sqrt u$ si y solo si $v^n> u$ (n impar o $u\\ge 0$)",
"Si $0 \\le a(^n\\sqrt u)$, entonces $v > a(^n\\sqrt u)$ si y solo si $v^n > a^nu$",
"Si $a\\ge 0$, entonces $^n\\sqrt v > a$ si y solo si $v > a^n$",
"u > v si y solo si $u^n > v^n$ (n impar, n>0)",
"u > v si y solo si $u^n > v^n$ (n > 0 y $0 \\le u$)",
"Si $a<0$, entonces $^2^n\\sqrt u > a$ si y solo si $u \\ge 0$"
},
{ /* root_ineq4 */
"$u \\ge v$ si y solo si $^n\\sqrt u \\ge ^n\\sqrt v$ (n impar)",
"$a \\ge u^2^n$ si y solo si $^2^n\\sqrt a \\ge |u|$",
"$a \\ge u^2^n$ si y solo si $-^2^n\\sqrt a \\le u \\le ^2^n\\sqrt a$",
"$u^2^n \\ge a$ si y solo si $|u| \\ge ^2^n\\sqrt a$, si $a\\ge 0$",
"$u^2^n \\ge a$ si y solo si $u \\le -^2^n\\sqrt a$ o $u \\ge ^2^n\\sqrt a$",
"$v \\ge ^2^n\\sqrt u$ si y solo si $0 \\le u \\le v^2^n$",
"$v \\ge ^n\\sqrt u$ si y solo si $v^n \\ge u$ (n impar o $u\\ge 0$)",
"$v \\ge a(^n\\sqrt u)$ si y solo si $v^n \\ge a^nu$, si $0 \\le a(^n\\sqrt u)$",
"$^n\\sqrt v \\ge a$ si y solo si $a^n \\le v$, si $a \\ge 0$",
"$u \\ge v$ si y solo si $u^n \\ge v^n$ (n impar, $n \\ge 0$)",
"$u \\ge v$ si y solo si $u^n \\ge v^n$ (n > 0 y $0 \\le u$)",
"$^2^n\\sqrt u \\ge a$ si y solo si $u \\ge 0$, si $a \\le 0$"
},
{ /* zero_ineq3 */
"Si u > 0, entonces u/v > 0 si y solo si v > 0",
"Cambio de $u/\\sqrt v > 0$ en uv > 0",
"u/v > 0 si y solo si uv > 0",
"Cambio de $0 > u/\\sqrt v$ en 0 > uv",
"0 > u/v si y solo si 0 > uv",
"$0 > ax \\pm b$ si y solo si $0 > a(x\\pm b/a)$",
"0 > (x-a)(x-b) si y solo si a<x<b (siendo a<b)",
"(x-a)(x-b) > 0 si y solo si x<a o x>b (siendo a<b)"
},
{ /* zero_ineq4 */
"Si $u \\ge 0$, entonces $u/v \\ge 0$ si y solo si $v \\ge 0$",
"$u/\\sqrt v \\ge 0$ si y solo si $uv \\ge 0$",
"$u/v \\ge 0$ si y solo si uv > 0 o u = 0",
"$0 \\ge u/\\sqrt v$ si y solo si $0 \\ge uv$",
"$0 \\ge u/v$ si y solo si 0 > uv o u = 0",
"$0 \\ge ax \\pm b$ si y solo si $0 \\ge a(x\\pm b/a)$",
"Si $a\\le b$, entonces $0 \\ge (x-a)(x-b)$ si y solo si $a\\le x\\le b$",
"Si $a\\le b$, entonces $(x-a)(x-b)\\ge 0$ si y solo si $x\\le a$ o $b\\le x$",
},
{ /* binomial_theorem */
"Desarrollo aplicando la f�rmula del binomio", //expand by binomial theorem
"F�rmula del binomio (n k)",
"$$binomial(n,k) = factorial(n)/ factorial(k) * factorial(n-k)$$",
"n! = n(n-1)(n-2)...1",
"C�lculo del factorial",
arithstr,
"C�lculo del coeficiente binomial",
"Desarrollo del t�rmino bajo la $\\sum $",
"Evaluaci�n de la $\\sum $ calculada como racional",
"n! = n (n-1)!",
"n!/n = (n-1)!",
"n!/(n-1)! = n",
"n!/k! = n(n-1)...(n-k+1)",
"n/n! = 1/(n-1)!",
"(n-1)!/n! = 1/n",
"k!/n! =1/(n(n-1)...(n-k+1))"
},
{ /* factor_expansion */
"a^3+3a^2b+3ab^2+b^3 = (a+b)^3",
"a^3-3a^2b+3ab^2-b^3 = (a-b)^3",
"a^4+4a^3b+6a^2b^2+4ab^3+b^4 = (a+b)^4",
"a^4-4a^3b+6a^2b^2-4ab^3+b^4 = (a-b)^4",
"a^n+na^(n-1)b+...b^n = (a+b)^n",
"a^n-na^(n-1)b+...b^n = (a-b)^n"
},
{ /* sigma_notation */
"$\\sum $ 1 = n�mero de t�rminos",
"$\\sum $ -u = -$\\sum $ u",
"$\\sum $ cu = c$\\sum $ u (c constante)",
"$\\sum (u\\pm v) = \\sum u \\pm \\sum v$",
"$\\sum (u-v) = \\sum u - \\sum v$",
"Desarrollo de $\\sum $ empleando la notaci�n +",
"1+2+..+n = n(n+1)/2",
"$1^2+..+n^2 = n(n+1)(2n+1)/6$",
"$1+x+..+x^n=(1-x^(n+1))/(1-x)$",
"Extracci�n de los primeros t�rminos",
"Evaluaci�n de la $\\sum $ con un par�metro, en formato racional.",
"Evaluaci�n de la $\\sum $ con un par�metro, en formato decimal.",
"C�lculo num�rico de la $\\sum $ en formato racional.",
"C�lculo num�rico de la $\\sum $ en formato decimal.",
"Expresi�n de la sumatoria de t�rminos como un polinomio",
"Sumatoria telesc�pica",
},
{ /* advanced_sigma_notation */
"Desplazamiento de l�mites de indexaci�n de la sumatoria",
"Cambio de nombre de la variable de indexaci�n",
"$(\\sum u)(\\sum v) = \\sum \\sum uv$",
"Extracci�n del �ltimo t�rmino",
"$1^3+..+n^3 = n^2(n+1)^2/4$",
"$1^4+..+n^4=n(n+1)(2n+1)(3n^2+2n-1)/30$",
"$d/dx \\sum u = \\sum du/dx$",
"$\\sum du/dx = d/dx \\sum u$",
"$\\int \\sum u dx = \\sum \\int u dx$",
"$\\sum \\int u dx = \\int \\sum u dx$",
"$c\\sum u = \\sum cu$",
"$$sum(t,i,a,b)=sum(t,i,0,b)-sum(t,i,0,a-1)$$",
"$$sum(t,i,a,b)=sum(t,i,c,b)-sum(t,i,c,a-1)$$"
},
{ /* prove_by_induction */
"Selecci�n de la variable de inducci�n",
"Inicio de la demostraci�n de la propiedad del caso base, para el primer valor",
"Comienzo del paso de inducci�n",
"Aplicaci�n de la hip�tesis de inducci�n",
"Lo que se ten�a que probar"
},
{ /* trig_ineq */
"$|sin u| \\le 1$",
"$|cos u| \\le 1$",
"$sin u \\le u$ si $u\\ge 0$",
"$1 - u^2/2 \\le cos u$",
"$|arctan u| \\le \\pi /2$",
"$arctan u \\le u$ si $u\\ge 0$",
"$u \\le tan u$ si $0\\le u\\le \\pi /2$"
},
{ /* log_ineq1 */
"Extracci�n de la funci�n logaritmo natural de ambos miembros",
"Extracci�n del logaritmo de ambos miembros",
"u < ln v si y solo si e^u < v",
"ln u < v si y solo si u < e^v",
"u < log v si y solo si 10^u < v",
"log u < v si y solo si u < 10^v",
"u < v si y solo si ?^u < ?^v",
},
{ /* log_ineq2 */
"Extracci�n del logaritmo natural de ambos miembros",
"Extracci�n del logaritmo de ambos miembros",
"$u \\le ln v$ si y solo si $e^u \\le v$",
"$ln u \\le v$ si y solo si $u \\le e^v$",
"$u \\le log v$ si y solo si $10^u \\le v$",
"$log u \\le v$ si y solo si $u \\le 10^v$",
"$u ? v$ si y solo si $?^u \\le ?^v$", /* takes arg in menu mode */
},
{ /* log_ineq3 */
"Extracci�n del logaritmo natural de ambos miembros",
"Extracci�n del logaritmo de ambos miembros",
"ln u > v si y solo si si u > e^v",
"u > ln v si y solo si si e^u > v",
"log u > v si y solo si si u > 10^v",
"u > log v si y solo si si 10^u > v",
"u > v si y solo si si ?^u > ?^v",
},
{ /* log_ineq4 */
"Extracci�n del logaritmo natural de ambos miembros",
"Extracci�n del logaritmo de ambos miembros",
"$ln u \\ge v$ si y solo si $u \\ge e^v$",
"$u \\ge ln v$ si y solo si $e^u \\ge v$",
"$log u \\ge v$ si y solo si $u \\ge 10^v$",
"$u \\ge log v$ si y solo si $10^u \\ge v$",
"$u \\ge v$ si y solo si $?^u \\ge ?^v$", /* takes arg in menu mode */
"Exponenciales dominantes de los polinomios",
"Funciones algebraicas dominantes de los logaritmos",
},
{ /* logarithms_base10 */
"$10^(log a) = a$",
"$log 10^n = n$ ($n$ real)",
"log 1 = 0",
"log 10 = 1",
"$log a = (ln a)/(ln 10)$",
"u^v = 10^(v log u)",
"Factorizaci�n completa del n�mero",
"Factorizaci�n de potencias de 10",
"10^(n log a) = a^n",
"log(a/b) = -log(b/a)",
"log(b,a/c) = -log(b,c/a)",
},
{ /* logarithms */
"$log a^n = n log a$",
"$log ab = log a + log b$",
"$log 1/a = -log a$",
"$log a/b = log a - log b$",
"$log a + log b = log ab$",
"$log a - log b = log a/b$",
"$log a + log b - log c =log ab/c$",
"$n log a = log a^n (n real)$",
"$log \\sqrt a = \\onehalf log a$",
"$log ^n\\sqrt a = (1/n) log a$",
"log 1 = 0",
"Factorizaci�n completa del n�mero",
"Factorizaci�n de potencias de base",
"$log u = (1/?) log u^?$",
"Evaluaci�n num�rica del logaritmo",
"$log a = (ln a)/(ln 10)$"
},
{ /* logarithms_base_e */
"e^(ln a) = a",
"ln e = 1",
"ln 1 = 0",
"ln e^n = n (n real)",
"u^v = e^(v ln u)",
"e^((ln c) a) = c^a"
},
{ /* natural_logarithms */
"ln a^n = n ln a",
"$ln ab = ln a + ln b$",
"ln 1/a = -ln a",
"$ln a/b = ln a - ln b$",
"ln 1 = 0",
"Factorizaci�n completa del n�mero",
"$ln a + ln b = ln ab$",
"$ln a - ln b = ln a/b$",
"$ln a + ln b - ln c = ln (ab/c)$",
"$n ln a = ln a^n (n real)$",
"$ln \\sqrt a = \\onehalf ln a$",
"$ln ^n\\sqrt a = (1/n) ln a$",
"ln u = (1/?) ln u^?", /* user supplies exponent; needed for diff(ln x,x) from defn */
"C�lculo num�rico del logaritmo",
"ln(a/b) = -ln(b/a)"
},
{ /* reverse_trig */
"sin u cos v + cos u sin v = sin(u+v)",
"sin u cos v - cos u sin v = sin(u-v)",
"cos u cos v - sin u sin v = cos(u+v)",
"cos u cos v + sin u sin v = cos(u-v)",
"(sin u)/(1+cos u) = tan(u/2)",
"(1-cos u)/sin u = tan(u/2)",
"(1+cos u)/(sin u) = cot(u/2)",
"sin u/(1-cos u) = cot(u/2)",
"(tan u+tan v)/(1-tan u tan v) = tan(u+v)",
"(tan u-tan v)/(1+tan u tan v) = tan(u-v)",
"(cot u cot v-1)/(cot u+cot v) = cot(u+v)",
"(1+cot u cot v)/(cot v-cot u) = cot(u-v)",
"1-cos u = 2 sin^2(u/2)"
},
{ /* complex_polar_form */
"Formato polar",
"$r e^(i\\theta ) = r (cos \\theta + i sin \\theta )$",
"$|e^(i\\theta )| = 1$",
"$|Re^(i\\theta )|=R$ si $R\\ge 0$",
"$|Re^(i\\theta )| = |R|$",
"$-a = ae^(\\pi i)$",
"$^n\\sqrt (-a) = e^(\\pi i/n) ^n\\sqrt a si a\\ge 0$",
"a/(ce^(ti)) = ae^(-ti)/c",
"Teorema de De Moivre",
"Sustituci�n de enteros espec�ficos"
},
{ /* logs_to_any_base */
"b^(log(b,a)) = a",
"b^(n log(b,a)) = a^n",
"log(b,b) = 1",
"log(b,b^n) = n",
"log xy = log x + log y",
"log (1/x) = -log x",
"log x/y = log x-log y",
"log(b,1) = 0",
"Factorizaci�n de la base del logaritmo: log(4,x)=log(2^2,x)",
"log(b^n,x) = (1/n) log (b,x)",
"log x^n = n log x",
"Factorizaci�n de potencias de la base del logaritmo",
"log x + log y = log xy",
"log x - log y = log x/y",
"log x + log y - log z =log xy/z",
"n log x = log x^n (n real)"
},
{ /* change_base */
"log(b,x) = ln x / ln b",
"log(b,x) = log x / log b",
"log(b,x) = log(a,x) / log(a,b)",
"log(b^n,x) = (1/n) log (b,x)",
"log(10,x) = log x",
"log(e,x) = ln x",
"log x = ln x / ln 10",
"ln x = log x / log e",
"u^v = b^(v log(b,u))"
},
{ /* evaluate_trig_function */
"sin 0 = 0",
"cos 0 = 1",
"tan 0 = 0",
"$sin k\\pi = 0$", /* logically, these are needed to prove sin(x+2�)=sin x */
"$cos 2k\\pi = 1$", /* They have to be proved separately by induction */
"$tan k\\pi = 0$",
"Determinaci�n en grados del �ngulo de $[0, 360[$ igual en m�dulo $360\\deg $",
"Determinaci�n en grados del �ngulo de $[0, 2\\pi [$ igual en m�dulo $2\\pi $",
"�ngulo m�ltiplo de $90\\deg $",
"Empleo de un semi-tri�ngulo equil�tero de lados de longitudes 1-2-$\\sqrt 3$",
"Empleo de un tri�ngulo rect�ngulo is�sceles de lados de longitudes 1-1-$\\sqrt 2$",
"Conversi�n de radianes a grados",
"Conversi�n de grados a radianes",
"�ngulo = $a 30\\deg + b 45\\deg $ etc.",
"An�lisis Num�rico"
},
{ /* basic_trig */
"tan u = sin u / cos u",
"cot u = 1 / tan u",
"cot u = cos u / sin u",
"sec u = 1 / cos u",
"csc u = 1 / sin u",
"sin u / cos u = tan u",
"cos u / sin u = cot u"
},
{ /* trig_reciprocals */
"1 / sin u = csc u",
"1 / cos u = sec u",
"1 / tan u = cot u",
"1 / tan u = cos u / sin u",
"1 / cot u = tan u",
"1 / cot u = sin u / cos u",
"1 / sec u = cos u",
"1 / csc u = sin u",
"sin u = 1 / csc u",
"cos u = 1 / sec u",
"tan u = 1 / cot u"
},
{ /* trig_squares */
"$sin^2 u + cos^2 u = 1$",
"$1 - sin^2 u = cos^2 u$",
"$1 - cos^2 u = sin^2 u$",
"$sin^2 u = 1 - cos^2 u$",
"$cos^2 u = 1 - sin^2 u$",
"$sec^2 u - tan^2 u = 1$",
"$tan^2 u + 1 = sec^2 u$",
"$sec^2 u - 1 = tan^2 u$",
"$sec^2 u = tan^2 u + 1$",
"$tan^2 u = sec^2 u - 1$",
"$sin^(2n+1) u = sin u (1-cos^2 u)^n$",
"$cos^(2n+1) u = cos u (1-sin^2 u)^n$",
"$tan^(2n+1) u = tan u (sec^2 u-1)^n$",
"$sec^(2n+1) u = sec u (tan^2 u+1)^n$",
"(1-cos t)^n(1+cos t)^n = sin^(2n) t",
"(1-sin t)^n(1+sin t)^n = cos^(2n) t"
},
{ /* csc_and_cot_identities */
"$csc^2 u - cot^2 u = 1$",
"$cot^2 u + 1 = csc^2 u$",
"$csc^2 u - 1 = cot^2 u$",
"$csc^2 u = cot^2 u + 1$",
"$cot^2 u = csc^2 u - 1$",
"$csc(\\pi /2-\\theta ) = sec \\theta $",
"$cot(\\pi /2-\\theta ) = tan \\theta $",
"$cot^(2n+1) u = cot u (csc^2 u-1)^n$",
"$csc^(2n+1) u = csc u (cot^2 u+1)^n$"
},
{ /* trig_sum */
"sin(u+v)= sin u cos v + cos u sin v",
"sin(u-v)= sin u cos v - cos u sin v",
"cos(u+v)= cos u cos v - sin u sin v",
"cos(u-v)= cos u cos v + sin u sin v",
"tan(u+v)=(tan u+tan v)/(1-tan u tan v)",
"tan(u-v)=(tan u-tan v)/(1+tan u tan v)",
"cot(u+v)=(cot u cot v-1)/(cot u+cot v)",
"cot(u-v)=(1+cot u cot v)/(cot v-cot u)"
},
{ /* double_angle */
"$sin 2\\theta = 2 sin \\theta cos \\theta $",
"$cos 2\\theta = cos^2 \\theta - sin^2 \\theta $",
"$cos 2\\theta = 1 - 2 sin^2 \\theta $",
"$cos 2\\theta = 2 cos^2 \\theta - 1$",
"$cos 2\\theta + 1 = 2cos^2 \\theta $",
"$cos 2\\theta - 1 = - 2 sin^2 \\theta $",
"$tan 2\\theta = 2 tan \\theta /(1 - tan^2 \\theta )$",
"$cot 2\\theta = (cot^2 \\theta -1) / (2 cot \\theta )$",
"$sin \\theta cos \\theta = \\onehalf sin 2\\theta $",
"$2 sin \\theta cos \\theta = sin 2\\theta $",
"$cos^2 \\theta - sin^2 \\theta = cos 2\\theta $",
"$1 - 2 sin^2 \\theta = cos 2\\theta $",
"$2 cos^2 \\theta - 1 = cos 2\\theta $"
},
{ /* multiple_angles */
"$n\\theta = (n-1)\\theta + \\theta $",
"$n\\theta = ?\\theta +(n-?)\\theta $",
"$sin 3\\theta = 3 sin \\theta - 4 sin^3 \\theta $",
"$cos 3\\theta = -3 cos \\theta + 4 cos^3 \\theta $",
"Desarrollo de $sin n\\theta $ en $sin \\theta $, $cos \\theta $",
"Desarrollo de $cos n\\theta $ en $sin \\theta $, $cos \\theta $"
},
{ /* verify_identities */
"Multiplicaci�n cruzada",
"Intercambio de miembros",
"Pasar ? de izquierda a derecha",
"Pasar ? de derecha a izquierda",
"Suma de ? en ambos miembros",
"Resta de ? en ambos miembros",
"Multiplicaci�n de ambos miembros por ?",
"Eliminaci�n de un t�rmino de ambos miembros",
"Potencia a la que se elevan ambos miembros",
"Extracci�n de la ra�z cuadrada de ambos miembros",
"Extracci�n de la ra�z de ambos miembros",
"Aplicaci�n de la funci�n a ambos miembros",
arithstr,
"Verificaci�n num�rica",
"Sustituci�n v�a un cambio de variable de la forma u = ?",
},
{ /* solve_by_30_60_90 */
"$sin(u)=1/2$ si y solo si $u=\\pi /6$ o $5\\pi /6+2n\\pi $",
"$sin(u)=-1/2$ si y solo si $u=-\\pi /6$ o $-5\\pi /6+2n\\pi $",
"$sin(u)=\\sqrt 3/2$ si y solo si $u=\\pi /3$ o $2\\pi /3+2n\\pi $",
"$sin(u)=-\\sqrt 3/2$ si y solo si $4u=-\\pi /3$ o $-2\\pi /3+2n\\pi $",
"$cos(u)=\\sqrt 3/2$ si y solo si $u=\\pm \\pi /6 + 2n\\pi $",
"$cos(u)=-\\sqrt 3/2$ si y solo si $u=\\pm 5\\pi /6 + 2n\\pi $",
"$cos(u)=1/2$ si y solo si $u=\\pm \\pi /3+2n\\pi $",
"$cos(u)=-1/2$ si y solo si $u=\\pm 2\\pi /3+2n\\pi $",
"$tan(u)=1/\\sqrt 3$ si y solo si $u= \\pi /6 + n\\pi $",
"$tan(u)=-1/\\sqrt 3$ si y solo si $u= -\\pi /6 + n\\pi $",
"$tan(u)=\\sqrt 3$ si y solo si $u= \\pi /3 + n\\pi $",
"$tan(u)=-\\sqrt 3$ si y solo si $u= 2\\pi /3 + n\\pi $",
},
{ /* solve_by_45_45_90 */
"$sin u = 1/\\sqrt 2$ si $u=\\pi /4$ o $3\\pi /4 + 2n\\pi $",
"$sin u=-1/\\sqrt 2$ si $u=5\\pi /4$ o $7\\pi /4 + 2n\\pi $2",
"$cos u = 1/\\sqrt 2$ si $u=\\pi /4$ o $7\\pi /4 + 2n\\pi $",
"$cos u=-1/\\sqrt 2$ si $u=3\\pi /4$ o $5\\pi /4 + 2n\\pi $",
"tan u = 1 si $u= \\pi /4$ o $5\\pi /4 + 2n\\pi $",
"tan u = -1 si $u=3\\pi /4$ o $7\\pi /4 + 2n\\pi $",
},
{ /* zeroes_of_trig_functions */
"sin u = 0 si y solo si $u = n\\pi $",
"sin u = 1 si y solo si $u = \\pi /2+2n\\pi $",
"sin u = -1 si y solo si $u = 3\\pi /2+2n\\pi $",
"cos u = 0 si y solo si $u = (2n+1)\\pi /2$",
"cos u = 1 si y solo si $u = 2n\\pi $",
"cos u = -1 si y solo si $u = (2n+1)\\pi $",
"tan u = 0 si y solo si sin u = 0",
"cot u = 0 si y solo si cos u = 0",
},
{ /* inverse_trig_functions */
"sin u=c si $u= (-1)^narcsin c+n\\pi $",
"sin u=c si $u=arcsin(c)+2n\\pi $ o $2n\\pi +\\pi -arcsin(c)$",
"cos u=c si $u=\\pm arccos c+2n\\pi $",
"tan u=c si $u=arctan c+n\\pi $", /* c not � i */
"C�lculo de arcsin en forma exacta",
"C�lculo de arccos en forma exacta",
"C�lculo de arctan en forma exacta",
"arccot x = arctan (1/x)",
"arcsec x = arccos (1/x)",
"arccsc x = arcsin (1/x)",
"arcsin(-x) = -arcsin x",
"$arccos(-x) = \\pi -arccos x$",
"arctan(-x) = -arctan x",
"Expresi�n de las soluciones en forma peri�dica",
"Si |c|>1, no existe u tal que sin u = c",
"Si |c|>1, no existe u tal que cos u = c",
},
{ /* invsimp */
"$tan(arcsin x) = x/\\sqrt (1-x^2)$",
"$tan(arccos x) = \\sqrt (1-x^2)/x$",
"tan(arctan x) = x",
"sin(arcsin x) = x",
"$sin(arccos x) = \\sqrt (1-x^2)$",
"$sin(arctan x) = x/\\sqrt (x^2+1)$",
"$cos(arcsin x) = \\sqrt (1-x^2)$",
"cos(arccos x) = x",
"$cos(arctan x) = 1/\\sqrt (x^2+1)$",
"$sec(arcsin x) = 1/\\sqrt (1-x^2)$",
"$sec(arccos x) = 1/x$",
"$sec(arctan x) = \\sqrt (x^2+1)$",
"$arctan(tan \\theta ) = \\theta $6 si $-\\pi /2\\le \\theta \\le \\pi /2$",
"$arcsin(sin \\theta ) = \\theta $ si $-\\pi /2\\le \\theta \\le \\pi /2$",
"$arccos(cos \\theta ) = \\theta $ si $0\\le \\theta \\le \\pi $",
"arctan(tan x) = x + c1"
},
{ /* adding_arctrig_functions */
"arcsin x + arccos x = $\\pi /2$",
"$arctan x + arctan 1/x = \\pi x/2|x|$",
#if 0 /* Perhaps add these later */
"$arcsin x \\pm arcsin y = arcsin[x\\sqrt (1-y^2)\\pm y\\sqrt (1-x^2)]$",
"$arccos x + arccos y = arccos[xy-\\sqrt ((1-x^2)(1-y^2))]$",
"$arccos x - arccos y = arccos[xy+\\sqrt ((1-x^2)(1-y^2))]$",
"$arctan x + arctan y = arctan[(x+y)/(1-xy)]$",
"$arctan x - arctan y = arctan[(x-y)/(1+xy)]$",
#endif
},
{ /* complementary_trig */
"$sin(\\pi /2-\\theta ) = cos \\theta $",
"$cos(\\pi /2-\\theta ) = sin \\theta $",
"$tan(\\pi /2-\\theta ) = cot \\theta $",
"$cot(\\pi /2-\\theta ) = tan \\theta $",
"$sec(\\pi /2-\\theta ) = csc \\theta $",
"$csc(\\pi /2-\\theta ) = sec \\theta $",
"$sin \\theta = cos(\\pi /2-\\theta )$",
"$cos \\theta = sin(\\pi /2-\\theta )$",
"$tan \\theta = cot(\\pi /2-\\theta )$",
"$cot \\theta = tan(\\pi /2-\\theta )$",
"$sec \\theta = csc(\\pi /2-\\theta )$",
"$csc \\theta = sec(\\pi /2-\\theta )$"
},
{ /* complementary_degrees */
"$sin(90\\deg -\\theta ) = cos \\theta $",
"$cos(90\\deg -\\theta ) = sin \\theta $",
"$tan(90\\deg -\\theta ) = cot \\theta $",
"$cot(90\\deg -\\theta ) = tan \\theta $",
"$sec(90\\deg -\\theta ) = csc \\theta $",
"$csc(90\\deg -\\theta ) = sec \\theta $",
"$sin \\theta = cos(90\\deg -\\theta )$",
"$cos \\theta = sin(90\\deg -\\theta )$",
"$tan \\theta = cot(90\\deg -\\theta )$",
"$cot \\theta = tan(90\\deg -\\theta )$",
"$sec \\theta = csc(90\\deg -\\theta )$",
"$csc \\theta = sec(90\\deg -\\theta )$",
"$a\\deg + b\\deg = (a+b)\\deg $",
"$ca\\deg = (ca)\\deg $",
"$a\\deg /c = (a/c)\\deg $"
},
{ /* trig_odd_and_even */
"sin(-u) = - sin u",
"cos(-u) = cos u",
"tan(-u) = - tan u",
"cot(-u) = - cot u",
"sec(-u) = sec u",
"csc(-u) = - csc u",
"$sin^2(-u) = sin^2 u$",
"$cos^2(-u) = cos^2 u$",
"$tan^2(-u) = tan^2 u$",
"$cot^2(-u) = cot^2 u$",
"$sec^2(-u) = sec^2 u$",
"$csc^2(-u) = csc^2 u$"
},
{ /* trig_periodic */
"$sin(u+2\\pi ) = sin u$",
"$cos(u+2\\pi ) = cos u$",
"$tan(u+\\pi ) = tan u$",
"$sec(u+2\\pi ) = sec u$",
"$csc(u+2\\pi ) = csc u$",
"$cot(u+\\pi ) = cot u$",
"$sin^2(u+\\pi ) = sin^2 u$",
"$cos^2(u+\\pi ) = cos^2 u$",
"$sec^2(u+\\pi ) = sec^2 u$",
"$csc^2(u+\\pi ) = csc^2 u$",
"$sin u = -sin(u-\\pi )$",
"$sin u = sin(\\pi -u)$",
"$cos u = -cos(u-\\pi )$",
"$cos u = -cos(\\pi -u)$"
},
{ /* half_angle_identities */
"$sin^2(\\theta /2) = (1-cos \\theta )/2$",
"$cos^2(\\theta /2) = (1+cos \\theta )/2$",
"$sin^2(\\theta ) = (1-cos 2\\theta )/2$",
"$cos^2(\\theta ) = (1+cos 2\\theta )/2$",
"$sin \\theta cos \\theta = \\onehalf sin 2\\theta $",
"$tan(\\theta /2) = (sin \\theta )/(1+cos \\theta )$",
"$tan(\\theta /2) = (1-cos \\theta )/sin \\theta $",
"$cot(\\theta /2) = (1+cos \\theta )/(sin \\theta )$",
"$cot(\\theta /2) = sin \\theta /(1-cos \\theta )$",
"$sin(\\theta /2) = \\sqrt ((1-cos \\theta )/2) si sin(\\theta /2)\\ge 0$",
"$sin(\\theta /2) = -\\sqrt ((1-cos \\theta )/2) si sin(\\theta /2)\\le 0$",
"$cos(\\theta /2) = \\sqrt ((1+cos \\theta )/2) si cos(\\theta /2)\\ge 0$",
"$cos(\\theta /2) = -\\sqrt ((1+cos \\theta )/2) si cos(\\theta /2)\\le 0$",
"$\\theta = 2(\\theta /2)$"
},
{ /* product_and_factor_identities */
"$sin x cos x = \\onehalf sin 2x$",
"$sin x cos y = \\onehalf [sin(x+y)+sin(x-y)]$",
"$cos x sin y = \\onehalf [sin(x+y)-sin(x-y)]$",
"$sin x sin y = \\onehalf [cos(x-y)-cos(x+y)]$",
"$cos x cos y = \\onehalf [cos(x+y)+cos(x-y)]$",
"$sin x + sin y = 2 sin \\onehalf (x+y) cos \\onehalf (x-y)$",
"$sin x - sin y = 2 sin \\onehalf (x-y) cos \\onehalf (x+y)$",
"$cos x + cos y = 2 cos \\onehalf (x+y) cos \\onehalf (x-y)$",
"$cos x - cos y = -2 sin \\onehalf (x+y) sin \\onehalf (x-y)$",
"Sustituci�n de u,v por expresiones en forma trigonom�trica"
},
{ /* limits */
"Experimentaci�n num�rica",
"$lim u\\pm v = lim u \\pm lim v$",
"$lim u-v = lim u - lim v$",
"lim(t\32a,c) = c (c constante)",
"lim(t\32a,t) = a",
"lim cu=c lim u (c constante)",
"lim -u = -lim u",
"lim uv = lim u lim v",
"$lim u^n = (lim u)^n$",
"lim c^v=c^(lim v) (c constante > 0)",
"lim u^v=(lim u)^(lim v)",
"$lim \\sqrt u=\\sqrt (lim u)$ si lim u>0",
"$lim ^n\\sqrt u = ^n\\sqrt (lim u)$ si n es impar",
"$lim ^n\\sqrt u = ^n\\sqrt (lim u)$ si lim u > 0",
"lim(t\32a,f(t))=f(a) (polinomial f)",
"lim |u| = |lim u|"
},
{ /* limits_of_quotients */
"lim cu/v = c lim u/v (c constante)",
"lim c/v = c/lim v (c constante)",
"lim u/v = lim u/lim v",
"Factorizaci�n de (x-a)^n en el estudio del l�mite cuando x tiende a a",
"L�mite de una funci�n racional",
"$a^n/b^n = (a/b)^n$",
"Racionalizaci�n de la fracci�n",
"Extracci�n de los l�mites finitos no nulos", /* lim uv = lim u lim v where lim u is finite nonzero */
"Factorizaci�n de una constante",
"Multiplicaci�n de numerador y denominador por ?",
"Divisi�n de numerador y denominador por ?",
"lim u/v = lim (u/?) / lim (v/?)",
"(ab+ac+d)/q = a(b+c)/q + d/q", /* limapartandfactor */
/* example : (sin x cos h + cos x sin h - sin x)/h */
},
{ /* quotients_of_roots */
"$\\sqrt a/b = \\sqrt (a/b^2)$ si b>0",
"$\\sqrt a/b = -\\sqrt (a/b^2)$ si b<0",
"$^n\\sqrt a/b = ^n\\sqrt (a/b^n)$ (b>0 o n impar)",
"$^n\\sqrt a/b = -^n\\sqrt (a/b^n)$ (b<0, n par)",
"$a/\\sqrt b = \\sqrt (a^2/b)$ si $a\\ge 0$",
"$a/\\sqrt b = -\\sqrt (a^2/b)$ si $a\\le 0$",
"$a/^n\\sqrt b = ^n\\sqrt (a^n/b)$ ($a\\ge 0$ o n impar)",
"$a/^n\\sqrt b = -^n\\sqrt (a^n/b)$ ($a\\le 0$, n par)"
},
{ /* lhopitalmenu */
"Regla de L'Hospital",
"C�lculo directo de la derivada en un paso",
"lim u ln v = lim (ln v)/(1/u)",
"$lim u (ln v)^n = lim (ln v)^n/(1/u)$",
"$lim x^(-n) u = lim u/x^n$",
"lim u e^x = lim u/e^(-x)",
"Pasar la funci�n trigonom�trica al denominador",
"lim ?v = lim v/(1/?)",
"Determinaci�n del denominador com�n y simplificaci�n del numerador"
},
{ /* special_limits */
"(sin t)/t \32 1 como t\32""0",
"(tan t)/t \32 1 como t\32""0",
"(1-cos t)/t \32 0 como t\32""0",
"$(1-cos t)/t^2\32""\\onehalf $ como t\32""0",
"lim(t\32""0,(1+t)^(1/t)) = e",
"$(ln(1\\pm t))/t \32 \\pm 1$ como t\32""0",
"(e^t-1)/t \32 1 como t\32""0",
"(e^(-t)-1)/t \32 -1 como t\32""0",
"$lim(t\32""0,t^nln |t|)=0 (n > 0)$",
"lim(t\32""0,cos(1/t))= no definido",
"lim(t\32""0,sin(1/t))= no definido",
"lim(t\32""0,tan(1/t))= no definido",
"lim(t\32""$\\pm \\infty $,cos t)= no definido",
"lim(t\32""$\\pm \\infty $,sin t)= no definido",
"lim(t\32""$\\pm \\infty $,tan t)= no definido"
},
{ /* hyper_limits */
"(sinh t)/t \32 1 como t\32""0",
"(tanh t)/t \32 1 como t\32""0",
"(cosh t - 1)/t \32 0 como t\32""0",
"(cosh t - 1)/t^2\32""1/2 como t\32""0",
},
{ /* advanced_limits */
"lim ln u=ln lim u (si lim u > 0)",
"lim f(u)=f(lim u), f continua",
"Cambio de variable en el l�mite", /* lim(t\32a,f(g(t)))=lim(u\32g(a),f(u)) */
"C�lculo directo del l�mite en un paso",
"lim u^v = lim e^(v ln u)",
"lim ?v = lim v/(1/?)",
"Tal dominio no permite la existencia del l�mite",
"lim u = e^(lim ln u)",
"Teorema del t�rmino finito: uv\32""0 si v\32""0 & $|u|\\le c$",
"$lim \\sqrt u-v=lim (\\sqrt u-v)(\\sqrt u+v)/(\\sqrt u+v)$",
"lim u/v = l�mite de los t�rminos m�s significativos",
"T�rmino m�s significativo: lim(u+a)=lim(u) si a/u\32""0",
"Sustituci�n de la suma por los t�rminos m�s significativos",
"f(no definido) = no definido",
"lim(e^u) = e^(lim u)",
"lim(ln u) = ln(lim u)"
},
{ /* logarithmic_limits */
"$lim(t\32""0+,t ln t) = 0$",
"$lim(t\32""0+,t^n ln t) = 0 si n\\ge 1$",
"$lim(t\32""0+,t (ln t)^n) = 0 si n\\ge 1$",
"$lim(t\32""0+,t^k (ln t)^n) = 0 si k,n\\ge 1$",
"$lim(t\32\\infty ,ln(t)/t) = 0$",
"$lim(t\32\\infty ,ln(t)^n/t) = 0 si n\\ge 1$",
"$lim(t\32\\infty ,ln(t)/t^n) = 0 si n\\ge 1$",
"$lim(t\32\\infty ,ln(t)^k/t^n) = 0 si k,n\\ge 1$",
"$lim(t\32\\infty ,t/ln(t)) = \\infty $",
"$lim(t\32\\infty ,t/ln(t)^n) = \\infty si n\\ge 1$",
"$lim(t\32\\infty ,t^n/ln(t)) = \\infty si n\\ge 1$",
"$lim(t\32\\infty ,t^n/ln(t)^k) = \\infty si k,n\\ge 1$"
},
{ /* limits_at_infinity */
"$lim(t\32\\infty ,1/t^n) = 0 si n\\ge 1$",
"$lim(t\32\\infty ,t^n) = \\infty si n\\ge 1$",
"$lim(t\32\\infty ,e^t) = \\infty $",
"$lim(t\32-\\infty ,e^t) = 0$",
"$lim(t\32\\infty ,ln t) = \\infty $",
"$lim(t\32\\infty ,\\sqrt t) = \\infty $",
"$lim(t\32\\infty ,^n\\sqrt t) = \\infty $",
"$lim(t\32\\pm \\infty ,arctan t) = \\pm \\pi /2$",
"$lim(t\32\\infty ,arccot t) = 0$",
"$lim(t\32-\\infty ,arccot t) = \\pi $",
"$lim(t\32\\pm \\infty ,tanh t) = \\pm 1$",
"$lim \\sqrt u-v=lim (\\sqrt u-v)(\\sqrt u+v)/\\sqrt u+v)$",
"lim sin u = sin(lim u)",
"lim cos u = cos(lim u)",
"Trasformaci�n de un l�mite en $\\infty $ en un l�mite en 0",
"lim u/v = l�mite de los t�rminos m�s significativos"
},
{ /* infinite_limits */
"$lim(1/u^2^n) = \\infty $ si u\32""0",
"lim(1/u^n) indefinido si u\32""0, n impar",
"lim(t\32a+,1/u^n) = $\\infty $ si u\32""0",
"lim(t\32a-,1/u^n)=-$\\infty $, u\32""0, n impar",
"lim u/v indefinido si lim v =0, lim u #0",
"lim(t\32""0+,ln t) = -$\\infty $",
"$lim(t\32(2n+1)\\pi /2\\pm ,tan t) = \\pm \\infty $",
"$lim(t\32n\\pi \\pm ,cot t) = \\pm \\infty $",
"$lim(t\32(2n+1)\\pi /2\\pm ,sec t) = \\pm \\infty $",
"$lim(t\32n\\pi \\pm ,csc t) = \\pm \\infty $",
"lim(uv) = lim(u/?) lim(?v)",
"lim(uv) = lim(?u) lim(v/?)"
},
{ /* infinities */
"$\\pm \\infty $/positivo = $\\pm \\infty $",
"no nulo/$\\pm \\infty $ = 0",
"positivo $\\times \\pm \\infty = \\pm \\infty $",
"$\\pm \\infty \\times \\infty = \\pm \\infty $",
"$\\pm \\infty $ + n�mero finito = $\\pm \\infty $",
"$\\infty + \\infty = \\infty $",
"$u^\\infty = \\infty $ si u > 1",
"$u^\\infty = 0$ si 0 < u < 1",
"$u^(-\\infty ) = 0$ si u > 1",
"$u^(-\\infty ) = \\infty $ si 0 < u < 1",
"$\\infty ^n = \\infty $ si n > 0",
"$\\infty - \\infty =$ indefinido"
},
{ /* zero_denom */
"$a/0+ = \\infty $ si $a>0$",
"$a/0- = -\\infty $ si $a>0$",
"a/0 = indefinido",
"$\\infty /0+ = \\infty $",
"$\\infty /0- = -\\infty $",
"$\\infty /0$ = no definido",
"$\\infty /0^2 = \\infty $",
"$\\infty /0^2^n = \\infty $",
"$a/0^2 = \\infty $ si $a > 0$",
"$a/0^2 = -\\infty $ si $a < 0$",
"$a/0^2^n = \\infty $ si $a > 0$",
"$a/0^2^n = -\\infty $ si $a < 0$"
},
{ /* more_infinities */
"$ln \\infty = log \\infty = \\infty $",
"$\\sqrt \\infty = \\infty $",
"$^n\\sqrt \\infty = \\infty $",
"$arctan \\pm \\infty = \\pm \\pi /2$",
"$arccot \\infty = 0$",
"$arccot -\\infty = \\pi $",
"$arcsec \\pm \\infty = \\pi /2$",
"$arccsc \\pm \\infty = 0$",
"L�mite trigonom�trico en $\\infty $ no definido",
"$cosh \\pm \\infty = \\infty $",
"$sinh \\pm \\infty = \\pm \\infty $",
"$tanh \\pm \\infty = \\pm 1$",
"$ln 0 = -\\infty $"
},
{ /* polynomial_derivs */
"Si c es una constante, dc/dx=0",
"dx/dx = 1",
"$d/dx (u \\pm v) = du/dx \\pm dv/dx$",
"d/dx (-u) = -du/dx",
"d/dx(cu)=c du/dx (c no dependiente de x)",
"d/dx x^n = n x^(n-1)",
"Derivaci�n de polinomio",
"f'(x) = d/dx f(x)"
},
{ /* derivatives */
"$$diff(f,x) = lim(h->0,(f(x+h)-f(x))/h)$$",
"Derivaci�n de polinomio",
"$d/dx (u \\pm v) = du/dx \\pm dv/dx$",
"d/dx (-u) = -du/dx",
"d/dx (cu) = c du/dx (c independiente de x)",
"d/dx (u/c)=(1/c)du/dx (c independiente de x)",
"d/dx x^n = n x^(n-1)",
"d/dx (uv) = u (dv/dx) + v (du/dx)",
"d/dx (1/v) = -(dv/dx)/v^2",
"d/dx (u/v)=[v(du/dx)-u(dv/dx)]/v^2",
"$d/dx \\sqrt x = 1/(2\\sqrt x)$",
"$d/dx ^n\\sqrt x = d/dx x^(1/n)$",
"$d/dx (c/x^n) = -nc/x^(n+1)$",
"d/dx |x| = x/|x|",
"f'(x) = d/dx f(x)"
},
{ /* dif_trig */
"d/dx sin x = cos x",
"d/dx cos x = - sin x",
"d/dx tan x = sec^2 x",
"d/dx sec x = sec x tan x",
"d/dx cot x = - csc^2 x",
"d/dx csc x = - csc x cot x"
},
{ /* dif_explog */
"d/dx e^x = e^x",
"d/dx c^x = (ln c) c^x, c constante",
"d/dx u^v= (d/dx) e^(v ln u)",
"d/dx ln x = 1/x",
"d/dx ln |x| = 1/x",
"dy/dx = y (d/dx) ln y",
"d/dx e^u = e^u du/dx",
"d/dx c^u=(ln c)c^u du/dx, c constante",
"d/dx ln u = (1/u)(du/dx)",
"d/dx ln |u| = (1/u) du/dx",
"d/dx ln(cos x) = -tan x",
"d/dx ln(sin x) = cot x"
},
{ /* dif_inverse_trig */
"$d/dx arctan x = 1/(1+x^2)$",
"$d/dx arcsin x = 1/\\sqrt (1-x^2)$",
"$d/dx arccos x = -1/\\sqrt (1-x^2)$",
"$d/dx arccot x = -1/(1+x^2)$",
"$d/dx arcsec x = 1/(|x|\\sqrt (x^2-1))$",
"$d/dx arccsc x = -1/(|x|\\sqrt (x^2-1))$",
"$d/dx arctan u = (du/dx)/(1+u^2)$",
"$d/dx arcsin u = (du/dx)/\\sqrt (1-u^2)$",
"$d/dx arccos u = -(du/dx)/\\sqrt (1-u^2)$",
"$d/dx arccot u = -(du/dx)/(1+u^2)$",
"$d/dx arcsec u=(du/dx)/(|u|\\sqrt (u^2-1))$",
"$d/dx arccsc u=-(du/dx)/(|u|\\sqrt (u^2-1))$"
},
{ /* chain_rule */
"d/dx u^n = nu^(n-1) du/dx",
"$d/dx \\sqrt u = (du/dx)/(2\\sqrt u)$",
"d/dx sin u = (cos u) du/dx",
"d/dx cos u = -(sin u) du/dx",
"$d/dx tan u = (sec^2 u) du/dx$",
"d/dx sec u=(sec u tan u) du/dx",
"$d/dx cot u = -(csc^2 u) du/dx$",
"d/dx csc u=-(csc u cot u) du/dx",
"d/dx |u| = (u du/dx)/|u|",
"d/dx f(u) = f'(u) du/dx",
"Cambio de variable en una sustituci�n de la forma $u = ?$",
"Eliminaci�n de una variable definida"
},
{ /* minima_and_maxima */
"Experimentaci�n num�rica",
"Estudio de los puntos de anulaci�n de la derivada, en que f'(x)=0",
"Estudio de los extremos del intervalo de estudio",
"Estudio de puntos de no diferenciaci�n, en que f'(x) no est� definida",
"Determinaci�n de l�mites de la funci�n en los extremos abiertos del intervalo",
"Rechazo de los puntos fuera del intervalo de estudio",
"Cuadro en que se relacione el valor decimal aproximado de la funci�n para cada punto propuesto",
"Cuadro en que se relacione el valor exacto de la funci�n para cada punto propuesto",
"Selecci�n del l�mite superior",
"Selecci�n del l�mite inferior",
"Evaluaci�n directa de la derivada calculada en un solo paso",
"Resoluci�n de una ecuaci�n simple",
"Determinaci�n del l�mite en un solo paso",
"Eliminaci�n de los par�metros enteros",
"La funci�n es constante",
// "eliminate open endpoints" // as of 5.24.13 no operation corresponds to this text. Should there be one?
},
{ /* implicit_diff */
"C�lculo de la derivada",
"Simplificaci�n",
"Resoluci�n de una ecuaci�n simple",
},
{ /* related_rates */
"Diferenciaci�n de la ecuaci�n",
"C�lculo de la derivada en un paso",
"Eliminaci�n de la derivada por sustituci�n, aplicando un cambio de variable",
"Resoluci�n de una ecuaci�n simple"
},
{ /* simplify */
"Simplificaci�n de sumas y productos",
"Eliminaci�n de fracciones compuestas",
"Determinaci�n del denominador com�n y simplificaci�n",
"Factorizaci�n del t�rmino com�n",
"Factorizaci�n del la expresi�n (no entera)",
"Desarrollo de productos y simplificaci�n", /* meaning either collect ou cancel ou both */
"Determinaci�n del factor com�n en u/v",
"Resoluci�n de una ecuaci�n simple",
"Expresar como poliomio (en ?)",
"Desarrollo como polinomio",
"Establecer el coeficiente principal como 1",
"$x^(1/2) = \\sqrt x$", /* backtosqrts */
"Conversi�n de potencias de exponentes racionales en ra�ces",
"Conversi�n de ra�ces en potencias de exponentes racionales",
},
{ /* higher_derivatives */
"u=v => du/dx = dv/dx",
"$d^2u/dx^2 = (d/dx)(du/dx)$",
"$d^nu/dx^n= d/dx d^(n-1)u/dx^(n-1)$",
"$d/dx du/dx = d^2u/dx^2$",
"$d/dx d^nu/dx^n = d^(n+1)/dx^(n+1)$",
"C�lculo de la derivada directamente, en un paso",
"Evaluaci�n num�rica, calculada en un punto",
},
{ /* basic_integration */
"$\\int 1 dt = t$",
"$\\int c dt = ct$ (c constante)",
"$\\int t dt = t^2/2$",
"$\\int cu dt = c\\int u dt$ (c constante)",
"$\\int (-u)dt = -\\int u dt$",
"$\\int u+v dt = \\int u dt + \\int v dt$",
"$\\int u-v dt = \\int u dt - \\int v dt$",
"$\\int au\\pm bv dt = a\\int u dt \\pm b\\int v dt$",
"$\\int t^n dt=t^(n+1)/(n+1) (n # -1)$",
"$\\int 1/t^(n+1) dt= -1/(nt^n) (n # 0)$",
"Integraci�n de polinomios hacia sus primitivas",
"$\\int (1/t) dt = ln |t|$",
"$\\int 1/(t\\pm a) dt = ln |t\\pm a|$",
"Desarrollo de productos del integrando",
"Desarrollo de $(a+b)^n$ en el integrando",
"$\\int |t| dt = t|t|/2$"
},
{ /* trig_integration */
"$\\int sin t dt = -cos t$",
"$\\int cos t dt = sin t$",
"$\\int tan t dt = -ln |cos t|$",
"$\\int cot t dt = ln |sin t|$",
"$\\int sec t dt = ln |sec t + tan t|$",
"$\\int csc t dt = ln |csc t - cot t|$",
"$\\int sec^2 t dt = tan t$",
"$\\int csc^2 t dt = -cot t$",
"$\\int tan^2 t dt = tan t - t$",
"$\\int cot^2 t dt = -cot t - t$",
"$\\int sec t tan t dt = sec t$",
"$\\int csc t cot t dt = -csc t$"
},
{ /* trig_integration2 */
"$\\int sin ct dt = -(1/c) cos ct$",
"$\\int cos ct dt = (1/c) sin ct$",
"$\\int tan ct dt = -(1/c) ln |cos ct|$",
"$\\int cot ct dt = (1/c) ln |sin ct|$",
"$\\int sec ct dt = (1/c) ln |sec ct + tan ct|$",
"$\\int csc ct dt = (1/c) ln |csc ct - cot ct|$",
"$\\int sec^2 ct dt = (1/c) tan ct$",
"$\\int csc^2 ct dt = -(1/c) cot ct$",
"$\\int tan^2 ct dt = (1/c) tan ct - t$",
"$\\int cot^2 ct dt = -(1/c) cot ct - t$",
"$\\int sec ct tan ct dt = (1/c) sec ct$",
"$\\int csc ct cot ct dt = -(1/c) csc ct$"
},
{ /* integrate_exp */
"$\\int e^t dt = e^t$",
"$\\int e^ct dt =(1/c) e^(ct)$",
"$\\int e^(-t)dt = -e^(-t)$",
"$\\int e^(-ct)dt = -(1/c) e^(-ct)$",
"$\\int e^(t/c)dt = c e^(t/c)$",
"$\\int c^t dt = (1/ln c) c^t$",
"$\\int u^v dt = \\int (e^(v ln u) dt$",
"$\\int ln t = t ln t - t$",
"$$integral(e^(-t^2),t) = sqrt(pi)/2 Erf(t)$$"
},
{ /* integrate_by_substitution */
"Selecci�n de la sustituci�n u = ?",
"Elecci�n a cargo de MathXpert, de la funci�n u para una sustituci�n por cambio de variable",
"Diferenciaci�n de la ecuaci�n",
"C�lculo directo de la derivada, en un paso",
"Visualizaci�n de la integral, recuperada",
"Integrando = $f(u) \\times du/dx$",
"$\\int f(u) (du/dx) dx = \\int f(u) du$",
"Eliminaci�n de una variable que ya hubiera sido definida",
"Integraci�n por cambio de variable (u = ?)",
"Integraci�n por cambio de variable",
},
{ /* integrate_by_parts */
"$\\int u dv = uv - \\int v du (u = ?)$",
"$\\int u dv = uv - \\int v du$",
"La l�nea actual, de aqu� en adelante si considerar� como la original.",
"integral original a izquierda",
"C�lculo directo de la derivada, en un paso",
"Integraci�n por cambio de variable (u = ?)",
"Integraci�n por cambio de variable",
"C�lculo de la integral simple"
},
{ /* fundamental_theorem */
"$$integral(f'(x),x,a,b)=f(b)-f(a)$$",
"$$diff(integral(f(t),t,a,x),x) = f(x)$$"
},
{ /* definite_integration */
"$$eval(f(t),t,a,b) = f(b) - f(a)$$",
"$$eval(ln f(t),t,a,b) = ln(f(b)/f(a))$$",
"$$integral(u,t,a,b) = - integral(u,t,b,a)$$",
"$$integral(u,t,a,b) + integral(u,t,b,c) = integral(u,t,a,c)$$",
"$$integral(u,t,a,c) = integral(u,t,a,?) + integral(u,t,?,c)$$",
"Descomposici�n de la integral $\\int |f(t)| dt$ seg�n los ceros de f",
"Evaluaci�n de la integral por c�lculo num�rico con un par�metro.",
"Evaluaci�n de la integral por c�lculo num�rico",
"$$integral(u,t,a,a) = 0$$"
},
{ /* improper_integrals */
"$$integral(u,x,a,infinity) = lim(t->infinity,integral(u,x,a,t))$$",
"$$integral(u,x,-infinity,b) = lim(t->-infinity,integral(u,x,t,b))$$",
"$$integral(u,x,a,b) = lim(t->a+,integral(u,x,t,b))$$",
"$$integral(u,x,a,b) = lim(t->b-,integral(u,x,a,t))$$",
"El l�mite del integrando no tiende a 0 en $\\infty $",
"El l�mite del integrando no tiende a 0 en $-\\infty $",
},
{ /* oddandeven */
"$$integral(u,t,-a,a) = 0$$ (u impar)",
"$$integral(u,t,-a,a) = 2 integral(u,t,0,a)$$ (u par)"
},
{ /* trig_substitutions */
"$x = a sin \\theta {for \\sqrt (a^2-x^2)}$",
"$x = a tan \\theta {for \\sqrt (a^2+x^2)}$",
"$x = a sec \\theta {for \\sqrt (x^2-a^2)}$",
"$x = a sinh \\theta {for \\sqrt (a^2+x^2)}$",
"$x = a cosh \\theta {for \\sqrt (x^2-a^2)}$",
"$x = a tanh \\theta {for \\sqrt (a^2-x^2)}$",
"Definici�n de la funci�n inversa por sustituci�n, v�a cambio de variable, x = ?",
"C�lculo de la derivada",
"Integraci�n elemental directa, en un solo paso",
},
{ /* trigonometric_integrals */
"$sin^2 t = (1-cos 2t)/2$ en la integral",
"$cos^2 t = (1+cos 2t)/2$ en la integral",
"u=cos x tras haber aplicado $sin^2=1-cos^2$",
"u=sin x tras haber aplicado $cos^2=1-sin^2$",
"u=tan x tras haber aplicado $sec^2=1+tan^2$",
"u=cot x tras haber aplicado $csc^2=1+cot^2$",
"u=sec x tras haber aplicado $tan^2=sec^2-1$",
"u=csc x tras haber aplicado $cot^2=csc^2-1$",
"$tan^2 x = sec^2 x - 1$ en el integrando",
"$2cot^2 x = csc^2 x - 1$ en el integrando",
"Reducci�n de $\\int sec^n x dx$",
"Reducci�n de $\\int csc^n x dx$",
"u = tan(x/2) (Cambio de variable de Weierstrass)"
},
{ /* trigrationalize */
"Multiplicaci�n del numerador y del denominador por 1+cos x",
"Multiplicaci�n del numerador y del denominador por 1-cos x",
"Multiplicaci�n del numerador y del denominador por 1+sin x",
"Multiplicaci�n del numerador y del denominador por 1-sin x",
"Multiplicaci�n del numerador y del denominador por sin x+cos x",
"Multiplicaci�n del numerador y del denominador por cos x-sin x"
},
{ /* integrate_rational*/
"Divisi�n polinomial",
"Factorizaci�n del denominador (de ser simple)",
"Extracci�n del factor com�n en u/v",
"Factorizaci�n sin cuadrados",
"Factorizaci�n num�rica del polinomio",
"Descomposici�n en elementos simples o en fracciones parciales",
"Operaci�n para completar el cuadrado en formato can�nico",
"$\\int 1/(ct\\pm b) dt = (1/c) ln |ct\\pm b|$",
"$\\int 1/(ct\\pm b)^(n+1) dt = -1/nc(ct\\pm b)^n$",
"$\\int 1/(t^2+a^2)dt=(1/a)arctan(t/a)$",
"$\\int 1/(t^2-a^2)dt=(1/a)arccoth(t/a)$",
"$\\int 1/(t^2-a^2)dt=(1/2a)ln|(t-a)/(t+a)|$",
"$\\int 1/(a^2-t^2)dt=(1/a)arctanh(t/a)$",
"$\\int 1/(a^2-t^2)dt=(1/2a)ln|(t+a)/(a-t)|$"
},
{ /* integrate_sqrtdenom */
"Operaci�n para completar el cuadrado en formato can�nico",
"$\\int 1/\\sqrt (a^2-t^2)dt = arcsin(t/a)$",
"$\\int 1/\\sqrt (t^2\\pm a^2)dt)=ln|t+\\sqrt (t^2\\pm a^2)|$",
"$\\int 1/(t\\sqrt (t^2-a^2))dt=(1/a)arccos(t/a)$",
"Cambio de variable, dando lugar a una fracci�n racional",
},
{ /* integrate_arctrig */
"$\\int arcsin z dz = z arcsin z + \\sqrt (1-z^2)$",
"$\\int arccos z dz = z arccos z - \\sqrt (1-z^2)$",
"$\\int arctan z dz = z arctan z - (1/2)ln(1+z^2)$",
"$\\int arccot z dz = z arccot z + (1/2)ln(1+z^2)$",
"$\\int arccsc z dz = z arccsc z+ln(z + \\sqrt (z^2-1)) (z>0)$",
"$\\int arccsc z dz = z arccsc z-ln(z + \\sqrt (z^2-1)) (z<0)$",
"$\\int arcsec z dz = z arcsec z-ln(z + \\sqrt (z^2-1)) (z>0)$",
"$\\int arcsec z dz = z arcsec z+ln(z + \\sqrt (z^2-1)) (z<0)$"
},
{ /* simplify_calculus */
"Simplificaci�n",
"Eliminaci�n de fracciones compuestas",
"Denominador com�n y simplificaci�n",
"Factorizaci�n de la expresi�n (no completa)",
"Factor com�n de los t�rminos",
"Desarrollo de productos y simplificaci�n", /* meaning either collect o cancel o both */
"Factor com�n de u/v",
"Resoluci�n de una ecuaci�n simple",
"Evaluaci�n de una derivada en un solo paso",
"Evaluaci�n de un l�mite en un solo paso",
"Trasformaci�n de la integral por sustituci�n, gracias a un cambio de variable",
"Integraci�n simple, en un solo paso",
"Incorporaci�n del n�mero a la constante de la primitiva"
},
{ /* integrate_hyperbolic */
"$\\int sinh u du = cosh u$",
"$\\int cosh u du = sinh u$",
"$\\int tanh u du = ln cosh u$",
"$\\int coth u du = ln sinh u$",
"$\\int csch u du = ln tanh(u/2)$",
"$\\int sech u du = arctan (sinh u)$"
},
{ /* series_geom1 */
"$$1/(1-x) = sum(x^n,n,0,infinity)$$",
"$1/(1-x) = 1+x+x^2+...$",
"$1/(1-x) = 1+x+x^2+...x^n...$",
"$$1/(1+x) = sum((-1)^n x^n,n,0,infinity)$$",
"$1/(1+x) = 1-x+x^2+...$",
"$1/(1+x) = 1-x+x^2+...(-1)^nx^n...$",
"$$sum(x^n,n,0,infinity)=1/(1-x)$$",
"$1+x+x^2+... = 1/(1-x)$",
"$1+x+x^2+...x^n...= 1/(1-x)$",
"$$sum((-1)^n x^n,n,0,infinity) = 1/(1+x)$$",
"$1-x+x^2+... = 1/(1+x)$",
"$1-x+x^2+...(-1)^nx^n... = 1/(1+x)$"
},
{ /* series_geom2 */
"$$x/(1-x) = sum(x^n,n,1,infinity)$$",
"$x/(1-x) = x+x^2+x^3+...$",
"$x/(1-x) = x+x^2+...x^n...$",
"$$x/(1+x) = sum((-1)^(n+1) x^n,n,1,infinity)$$",
"$x/(1+x) = x-x^2+x^3+...$",
"$x/(1+x) = x-x^2+...(-1)^(n+1)x^n...$",
"$$sum(x^n,n,1,infinity)=x/(1-x)$$",
"$x+x^2+x^3+...=x/(1-x)$",
"$x+x^2+...x^n...=x/(1-x)$",
"$$sum((-1)^(n+1) x^n,n,1,infinity)=x/(1+x) $$",
"$x-x^2+x^3+...=x/(1+x) $",
"$x-x^2+...(-1)^(n+1)x^n...=x/(1+x) $"
},
{ /* series_geom3 */
"$$1/(1-x^k) = sum(x^(kn),n,0,infinity)$$",
"$$1/(1-x^k) = sum(x^(kn),n,0,infinity,-3)$$",
"$$1/(1-x^k) = sum(x^(kn),n,0,infinity,2)$$",
"$$x^m/(1-x^k) = sum(x^(kn+m),n,0,infinity)$$",
"$$x^m/(1-x^k) = sum(x^(kn+m),n,0,infinity,-3)$$",
"$$x^m/(1-x^k) = sum(x^(kn+m),n,0,infinity,2)$$",
"$$sum(x^(kn),n,0,infinity)=1/(1-x^k)$$",
"$$sum(x^(kn),n,0,infinity,-3)=1/(1-x^k)$$",
"$$sum(x^(kn),n,0,infinity,2)=1/(1-x^k)$$",
"$$sum(x^(m+kn),n,0,infinity)=x^m/(1-x^k)$$",
"$$sum(x^(m+kn),n,0,infinity,-3)=x^m/(1-x^k)$$",
"$$sum(x^(m+kn),n,0,infinity,2)=x^m/(1-x^k)$$"
},
{ /* series_geom4 */
"$$1/(1+x^k) = sum((-1)^n x^(kn),n,0,infinity)$$",
"$$1/(1+x^k) = sum((-1)^n x^(kn),n,0,infinity,-3)$$",
"$$1/(1+x^k) = sum((-1)^n x^(kn),n,0,infinity,2)$$",
"$$x^m/(1+x^k) = sum((-1)^n x^(kn+m),n,0,infinity)$$",
"$$x^m/(1+x^k) = sum((-1)^n x^(kn+m),n,0,infinity,-3)$$",
"$$x^m/(1+x^k) = sum((-1)^n x^(kn+m),n,0,infinity,2)$$",
"$$sum((-1)^nx^(kn),n,0,infinity)=1/(1+x^k)$$",
"$$sum((-1)^nx^(kn),n,0,infinity,-3)=1/(1+x^k)$$",
"$$sum((-1)^nx^(kn),n,0,infinity,2)=1/(1+x^k)$$",
"$$sum((-1)^nx^(m+kn),n,0,infinity)=x^m/(1+x^k)$$",
"$$sum((-1)^nx^(m+kn),n,0,infinity,-3)=x^m/(1+x^k)$$",
"$$sum((-1)^nx^(m+kn),n,0,infinity,2)=x^m/(1+x^k)$$"
},
{ /* series_geom5 */
"$$x^k/(1-x) = sum(x^n,n,k,infinity)$$",
"$$x^k/(1-x) = sum(x^n,n,k,infinity,-3)$$",
"$$x^k/(1-x) = sum(x^n,n,k,infinity,2)$$",
"$$x^k/(1+x) = sum((-1)^nx^n,n,k,infinity)$$",
"$$x^k/(1+x) = sum((-1)^nx^n,n,k,infinity,-3)$$",
"$$x^k/(1+x) = sum((-1)^nx^n,n,k,infinity,2)$$",
"$$sum(x^n,n,k,infinity) = x^k/(1-x)$$",
"$$sum(x^n,n,k,infinity,-3) = x^k/(1-x)$$",
"$$sum(x^n,n,k,infinity,2) = x^k/(1-x)$$",
"$$sum((-1)^nx^n,n,k,infinity) = x^k/(1+x)$$",
"$$sum((-1)^nx^n,n,k,infinity,-3) = x^k/(1+x)$$",
"$$sum((-1)^nx^n,n,k,infinity,2) = x^k/(1+x)$$"
},
{ /* series_ln */
"$$ln(1-x) = sum(x^n/n,n,1,infinity)$$",
"$$ln(1-x) = sum(x^n/n,n,1,infinity,-3)$$",
"$$ln(1-x) = sum(x^n/n,n,1,infinity,2)$$",
"$$ln(1+x) = sum((-1)^(n+1) x^n/n,n,1,infinity)$$",
"$$ln(1+x) = sum((-1)^(n+1) x^n/n,n,1,infinity,-3)$$",
"$$ln(1+x) = sum((-1)^(n+1) x^n/n,n,1,infinity,2)$$",
"$$sum(x^n/n,n,1,infinity) = ln(1-x)$$",
"$$sum(x^n/n,n,1,infinity,-3)=ln(1-x)$$",
"$$sum(x^n/n,n,1,infinity,2)=ln(1-x)$$",
"$$sum((-1)^(n+1) x^n/n,n,1,infinity)=ln(1+x)$$",
"$$sum((-1)^(n+1) x^n/n,n,1,infinity,-3)=ln(1+x)$$",
"$$sum((-1)^(n+1) x^n/n,n,1,infinity,2)=ln(1+x)$$"
},
{ /* series_trig */
"$$ sin x = sum( (-1)^n x^(2n+1)/(2n+1)!,n,0,infinity)$$",
"$sin x = x-x^3/3!+x^5/5!+...$",
"$sin x = x-x^3/3!+x^5/5!+...+ (-1)^nx^(2n+1)/(2n+1)!+...$",
"$$cos x = sum( (-1)^n x^(2n)/(2n)!,n,0,infinity)$$",
"$cos x = 1-\\onehalf x^2+x^4/4! + ...$",
"$cos x = 1-\\onehalf x^2+...+(-1)^nx^(2n)/(2n)!+...$",
"$$sum((-1)^n x^(2n+1)/(2n+1)!,n,0,infinity) = sin x$$",
"$x-x^3/3!+x^5/5!+... = sin x$",
"$x-x^3/3!+x^5/5!+...+ (-1)^nx^(2n+1)/(2n+1)!+... = sin x$",
"$$sum( (-1)^n x^(2n)/(2n)!,n,0,infinity) = cos x$$",
"$1-\\onehalf x^2+x^4/4! + ... = cos x$",
"$1-\\onehalf x^2+...+(-1)^nx^(2n)/(2n)!+... = cos x$"
},
{ /* series_exp */
"$$e^x = sum(x^n/n!,n,0,infinity)$$",
"$e^x = 1+x+x^2/2!+...$",
"$e^x = 1+x+...+x^n/n!...$",
"$$sum(x^n/n!,n,0,infinity)= e^x$$",
"$1+x+x^2/2!+ x^3/3!+... = e^x$",
"$1+x+...+x^n/n!... = e^x$",
"$$e^(-x) = sum((-x)^n x^n/n!,n,0,infinity)$$",
"$e^(-x) = 1-x+x^2/2!+...$",
"$e^(-x) = 1-x+...(-1)^nx^n/n!...$",
"$$sum((-1)^nx^n/n!,n,0,infinity)= e^(-x)$$",
"$1-x+x^2/2!+ x^3/3!+... = e^(-x)$",
"$1-x+...+(-1)^nx^n/n!... = e^(-x)$"
},
{ /* series_atan */
"$$arctan x = sum(x^(2n+1)/(2n+1),n,0,infinity)$$",
"$arctan x = x -x^3/3 + x^5/5 ...$",
"$arctan x = x -x^3/3 +...+ x^(2n+1)/(2n+1)+...$",
"$$sum(x^(2n+1)/(2n+1),n,0,infinity) = arctan x$$",
"$x -x^3/3 + x^5/5 ...=arctan x$",
"$x -x^3/3 +...+ x^(2n+1)/(2n+1)+...=arctan x$",
"$$(1+x)^alpha = sum(binomial(alpha,n) x^n,n,0,infinity)$$",
"$$(1+x)^alpha = sum(binomial(alpha,n) x^n,n,0,infinity,-3)$$",
"$$(1+x)^alpha = sum(binomial(alpha,n) x^n,n,0,infinity,2)$$",
"$$sum(binomial(alpha,n) x^n,n,0,infinity)= (1+x)^alpha$$",
"$$sum(binomial(alpha,n) x^n,n,0,infinity,-3)= (1+x)^alpha$$",
"$$sum(binomial(alpha,n) x^n,n,0,infinity,2)= (1+x)^alpha$$"
},
{ /* series_bernoulli */
"$$tan x = sum((-1)^(n-1) (2^(2n)(2^(2n)-1) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity)$$",
"$$tan x = sum((-1)^(n-1) (2^(2n)(2^(2n)-1) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity,-3)$$",
"$$tan x = sum((-1)^(n-1) (2^(2n)(2^(2n)-1) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity,2)$$",
"$$x cot x = sum((-1)^n (2^(2n) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity)$$",
"$$x cot x = sum((-1)^n (2^(2n) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity,-3)$$",
"$$x cot x = sum((-1)^n (2^(2n) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity,2)$$",
"$$x/(e^x-1) = 1-x/2 + sum(( bernoulli(2n))/((2n)!) x^(2n),n,1,infinity)$$",
"$$x/(e^x-1) = 1-x/2 + sum(( bernoulli(2n))/((2n)!) x^(2n),n,1,infinity,-3)$$",
"$$x/(e^x-1) = 1-x/2 + sum(( bernoulli(2n))/((2n)!) x^(2n),n,1,infinity,2)$$",
"$$sec x = sum( (-1)^n (eulernumber(2n))/((2n)!) x^(2n),n,1,infinity)$$",
"$$sec x = sum(( eulernumber(2n))/((2n)!) x^(2n),n,1,infinity,-3)$$",
"$$sec x = sum(( eulernumber(2n))/((2n)!) x^(2n),n,1,infinity,2)$$",
"$$zeta(s) = sum(1/n^s,n,1,infinity)$$",
"$$zeta(s) = sum(1/n^s,n,1,infinity,-3)$$",
"$$zeta(s) = sum(1/n^s,n,1,infinity,-2)$$",
"$$sum((-1)^n/n,n,1,infinity) = ln 2$$"
},
{ /* series_appearance */
"Expresi�n de la serie como $a_0 + a_1 + ...$",
"Expresi�n de la serie como $a_0 + a_1 + a_2 + ... $",
"Expresi�n de la serie usando ... y el t�rmino general",
"Expresi�n de la serie usando la notaci�n sigma",
"Inclusi�n de otro t�rmino antes de ...",
"Inclusi�n de ? t�rminos suplementarios antes de ...",
"Inclusi�n de los t�rminos tras el c�lculo de los factoriales",
"Sin evaluaci�n de factoriales en los t�rminos",
"Expresi�n de los coeficientes en formato decimal",
"Expresi�n no decimal de los coeficientes",
},
{ /* series_algebra */
"Serie telesc�pica",
"Multiplicaci�n de serie",
"Multiplicaci�n de serie de potencias",
"Divisi�n de serie de potencias por un polinomio",
"Divisi�n de polinomio por una serie de potencias",
"Divisi�n de la serie de potencias",
"Cuadrado de una serie",
"Cuadrado de una serie de potencias",
"Expresi�n de $(\\sum a_k x^k)^n$ como una serie",
"Suma de series",
"Resta de series",
},
{ /* series_manipulations */
"Extracci�n de los primeros t�rminos",
"Reducci�n del l�mite inferior despejando t�rminos",
"Suma de ? a la variable �ndice",
"Resta de ? de la variable �ndice",
"Cambio de nombre de la variable �ndice",
"$\\sum (u\\pm v) = \\sum u \\pm \\sum v$",
"Diferenciaci�n de la serie de potencias, t�rmino por t�rmino",
"$\\sum du/dx = d/dx \\sum u$",
"Integraci�n de la serie de potencias, t�rmino por t�rmino",
"$\\sum \\int u dx = \\int \\sum u dx$",
"C�lculo de la suma de los primeros t�rminos",
"$$u = integral(diff(u,x),x)$$",
"$$u = integral(diff(u,t),t,0,x) + u0$$",
"$$u = diff(integral(u,x),x)$$",
"Determinaci�n de la constante de integraci�n",
"$\\sum a_k = \\sum a_(2k) + \\sum a_(2k+1)$"
},
{ /* series_convergence_tests */
"$\\sum u$ diverge si u no tiende a cero",
"Regla de comparaci�n con una integral",
"Regla de D'Alembert",
"Regla de Cauchy",
"Regla de comparaci�n por convergencia",
"Regla de comparaci�n por divergencia",
"Regla de equivalentes",
"Regla de condensaci�n de Cauchy",
"Finalizaci�n del test de la divergencia",
"Finalizaci�n del test de comparaci�n con una integral",
"Finalizaci�n del test de la regla de D'Alembert",
"Finalizaci�n del test de la regla de Cauchy",
"Finalizaci�n del test de la comparaci�n",
"Finalizaci�n del test de la comparaci�n",
"Finalizaci�n del test de la comparaci�n por equivalentes",
"Finalizaci�n del test de la regla de condensaci�n de Cauchy",
},
{ /* series_convergence2 */
"Resultado positivo del test de comparaci�n",
"Resultado negativo del test de comparaci�n",
"$$sum(1/k,k,1,infinity) = infinity$$",
"$$sum(1/k^2,k,1,infinity) = pi^2/6$$",
"$$sum(1/k^s,k,1,infinity) = zeta(s)$$",
"$$zeta(2k) = (2^(2k-1) abs(bernoulli(2k)) pi^(2k))/factorial(2k)$$"
},
{ /* complex_functions */
"$ln(u+iv) = ln(re^(i\\theta ))$",
"$ln(re^(i\\theta ))=ln r + i\\theta (-\\pi <\\theta \\le \\pi )$",
"$ln i = i\\pi /2$",
"$ln(-1) = i\\pi $",
"$ln(-a) = ln a + i\\pi (a > 0)$",
"$cos \\theta = [e^(i\\theta ) + e^(-i\\theta )]/2$",
"$sin \\theta = [e^(i\\theta ) - e^(-i\\theta )]/2i$",
"$$sqrt(re^(i theta))=sqrt(r) e^(i theta/2)$$ $ (-\\pi < \\theta \\le \\pi )$",
"$$root(n,re^(i theta))=root(n,r) e^(i theta/n)$$ $ (-\\pi < \\theta \\le \\pi )$",
"$e^(i\\theta ) = cos \\theta + i sin \\theta $",
"$e^(x+iy) = e^x cos y + i e^x sin y$",
"$e^(i\\pi ) = -1$",
"$e^(-i\\pi ) = -1$",
"$e^(2n\\pi i) = 1$",
"$e^((2n\\pi + \\theta )i) = e^(i\\theta )$",
"$u^v = e^(v ln u)$"
},
{ /* complex_hyperbolic */
"sin(it) = i sinh t",
"cos(it) = cosh t",
"cosh(it) = cos t",
"sinh(it) = i sin t",
"tan(it) = i tanh t",
"cot(it) = -i coth t",
"tanh(it) = i tan t",
"coth(it) = -i cot t",
"cos t + i sin t = e^(it)",
"cos t - i sin t = e^(-it)",
"$[e^(i\\theta ) + e^(-i\\theta )]/2 = cos \\theta $",
"$[e^(i\\theta ) - e^(-i\\theta )]/2i = sin \\theta $",
"$e^(i\\theta ) + e^(-i\\theta ) = 2 cos \\theta $",
"$e^(i\\theta ) - e^(-i\\theta ) = 2i sin \\theta $"
},
{ /* hyperbolic_functions */
"cosh u = (e^u+e^(-u))/2",
"e^u + e^-u = 2 cosh u",
"sinh u = (e^u-e^(-u))/2",
"e^u-e^(-u) = 2 sinh u",
"[e^u + e^-u]/2 = cosh u",
"[e^u-e^(-u)]/2 = sinh u",
"cosh(-u) = cosh u",
"sinh(-u) = -sinh u",
"cosh u + sinh u = e^u",
"cosh u - sinh u = e^(-u)",
"cosh 0 = 1",
"sinh 0 = 0",
"e^x = cosh x + sinh x",
"e^(-x) = cosh x - sinh x"
},
{ /* hyperbolic2 */
"$sinh^2u + 1 = cosh^2 u$",
"$cosh^2 u - 1 = sinh^2u $",
"$cosh^2 u - sinh^2u = 1$",
"$cosh^2 u = sinh^2u + 1$",
"$sinh^2u = cosh^2 u - 1$",
"$1 - tan^2u = sech^2u$",
"$1 - sech^2u = tan^2u$"
},
{ /* more_hyperbolic */
"tanh u = sinh u / cosh u",
"sinh u / cosh u = tanh u",
"coth u = cosh u / sinh u",
"cosh u / sinh u = coth u",
"sech u = 1 / cosh u",
"1 / cosh u = sech u",
"csch u = 1 / sinh u",
"1 / sinh u = csch u",
"$tanh^2 u + sech^2 u = 1$",
"$tanh^2 u = 1 - sech^2 u$",
"$sech^2 u = 1 - tanh^2 u $",
"$sinh(u\\pm v)=sinh u cosh v \\pm cosh u sinh v$",
"$cosh(u\\pm v)=cosh u cosh v \\pm sinh u sinh v$",
"sinh 2u = 2 sinh u cosh u",
"$cosh 2u = cosh^2 u + sinh^2 u$",
"$tanh(ln u) = (1-u^2)/(1+u^2)$"
},
{ /* inverse_hyperbolic */
"$arcsinh x = ln(x + \\sqrt (x^2+1))$",
"$arccosh x = ln(x + \\sqrt (x^2-1))$",
"$arctanh x = (1/2) ln((1+x)/(1-x))$",
"$sinh(asinh x) = x$",
"$cosh(acosh x) = x$",
"$tanh(atanh x) = x$",
"$coth(acoth x) = x$",
"$sech(asech x) = x$",
"$csch(acsch x) = x$"
},
{ /* dif_hyperbolic */
"d/du sinh u = cosh u",
"d/du cosh u = sinh u",
"$d/du tanh u = sech^2 u$",
"$d/du coth u = -csch^2 u$",
"d/du sech u = -sech u tanh u",
"d/du csch u = -csch u coth u",
"d/du ln sinh u = coth u",
"d/du ln cosh u = tanh u"
},
{ /* dif_inversehyperbolic */
"$d/du arcsinh u = 1/\\sqrt (u^2+1)$",
"$d/du arccosh u = 1/\\sqrt (u^2-1)$",
"$d/du arctanh u = 1/(1-u^2)$",
"$d/du arccoth u = 1/(1-u^2)$",
"$d/du arcsech u= -1/(u\\sqrt (1-u^2))$",
"$d/du arccsch u= -1/(|u|\\sqrt (u^2+1))$"
},
{ /* sg_function1 */
"sg(x) = 1 si x > 0", /* sgpos */
"sg(x) = -1 si x < 0", /* sgneg */
"sg(0) = 0", /* sgzero */
"sg(-x) = -sg(x)", /* sgodd */
"-sg(x) = sg(-x)", /* sgodd2 */
"sg(x) = |x|/x (x no nulo)", /* sgabs1 */
"sg(x) = x/|x| (x no nulo)", /* sgabs2 */
"abs(x) = x sg(x)", /* abssg */
"$sg(x)^(2n) = 1$", /* also sg(x)^(even/odd) sgevenpower */
"sg(x)^(2n+1) = sg(x)", /* also sg(x)^odd/odd sgoddpower */
"1/sg(x) = sg(x)", /* sgrecip */
"d/dx sg(u) = 0 (u no nulo)", /* difsg */
"$\\int sg(x) = x sg(x)$", /* intsg */
"$\\int sg(u)v dx = sg(u)\\int v dx$ (u no nulo)", /* sgint */
"sg(x) = 1 siendo x > 0", /* sgassumepos */
"sg(x) = -1 siendo x < 0" /* sgassumeneg */
},
{ /* sg_function2 */
"$sg(au) = sg(u)$ si $a > 0$",
"$sg(au) = -sg(u)$ si a < 0",
"sg(au/b) = sg(u) si a/b > 0",
"sg(au/b) = - sg(u) si a/b < 0",
"sg(x^(2n+1)) = sg(x)",
"sg(1/u) = sg(u)",
"sg(c/u) = sg(u) si c > 0",
"u sg(u) = |u|",
"|u| sg(u) = u"
},
{ /* bessel_functions */
"d/dx J0(x) = -J1(x)",
"d/dx J1(x) = J0(x) - J1(x)/x",
"d/dx J(n,x)=J(n-1,x)-(n/x)J(n,x)",
"d/dx Y0(x) = -Y1(x)",
"d/dx Y1(x) = Y0(x) - Y1(x)/x",
"d/dx Y(n,x)=Y(n-1,x)-(n/x)Y(n,x)"
},
{ /* modified_bessel_functions */
"d/dx I0(x) = -I1(x)",
"d/dx I1(x) = I0(x) - I1(x)/x",
"d/dx I(n,x)=I(n-1,x)-(n/x)I(n,x)",
"d/dx K0(x) = -K1(x)",
"d/dx K1(x) = -K0(x) - K1(x)/x",
"d/dx K(n,x)= -K(n-1,x)-(n/x)K(n,x)"
},
{ /* functions_menu -- user-defined functions */
"" /* definitions of user-defined functions appear here. */
},
{"Desarrollo", /* automode_only, this menu never appears! */
"Multiplicaci�n de eliminarse" /* but model.c uses corresponding entries in optable */
},
{"Eliminaci�n de ra�ces cuadradas" /* automode_only2, also never appears */
},
{"" /* automode_only3, also never appears */
}
};
/*_______________________________________________________________*/
/* array of titles of the command menus */
static const char *const Spanish_menutitles[MAXMENUS] =
{
/* first the algebra menus */
"C�lculo Numerico",
"Expresi�n del n�mero en forma diversa",
"Aritm�tica Compleja",
"Simplificaci�n de la Sumatoria",
"Simplificaci�n del Producto",
"Desarrollo",
"Fracciones",
"Fracciones con signo",
"Fracciones compuestas",
"Denominadores comunes",
"Exponentes",
"Desarrollo de potencias",
"Exponentes negativos",
"Ra�ces cuadradas",
"Ra�ces cuadradas avanzada",
"Exponentes fraccionarios",
"Ra�ces e$n$-�sima",
"Ra�ces de ra�ces",
"Ra�ces y fracciones",
"N�meros complejos",
"Factorizaciones",
"Factorizaciones avanzada",
"Resoluci�n de ecuaciones",
"Ecuaciones cuadr�ticas",
"Estudio num�rico de ecuaciones",
"Ecuaciones avanzada",
"Ecuaciones c�bicas",
"Ecuaciones logar�tmicas o exponenciales",
"Regla de Cramer",
"Diversas ecuaciones lineales",
"Modo selecci�n exclusivo", /* This title is never shown */
"Ecuaciones lineales para selecci�n de los t�rminos", /* This title is never shown */
"Ecuaciones para sustituci�n",
"M�todos matriciales",
"M�todos matriciales avanzados",
"Valores absolutos",
"Inecuaciones con valores absolutos", /* absolute_value_ineq1 */
"Inecuaciones con valores absolutos", /* absolute_value_ineq2 */
"Inecuaciones estrictas", /* less_than */
"Inecuaciones estrictas", /* greater_than */
"Inecuaciones", /* less_than_or_equal */
"Inecuaciones", /* greater_than_or_equal */
"Inecuaciones con cuadrados",
"Inecuaciones con cuadrados",
"Inecuaciones con inversos",
"Inecuaciones con inversos",
"Inecuaciones con ra�ces y potencias",
"Inecuaciones con ra�ces y potencias",
"Inecuaciones--un miembro nulo",
"Inecuaciones--un miembro nulo",
"Inecuaciones con cuadrados", /* Now repeat the last six for > and GE */
"Inecuaciones con cuadrados",
"Inecuaciones con inversos",
"Inecuaciones con inversos",
"Inecuaciones con ra�ces y potencias",
"Inecuaciones con ra�ces y potencias",
"Inecuaciones--un miembro nulo",
"Inecuaciones--un miembro nulo",
"Teorema binomial",
"Factorizaciones de desarrollo binomial",
"Notaci�n sigma",
"Notaci�n sigma avanzada",
"Dimostraciones por inducci�n",
"Inecuaciones trigonom�tricas",
"Inecuaciones con logaritmos y potencias",
"Inecuaciones con logaritmos y potencias",
"Inecuaciones con logaritmos y potencias",
"Inecuaciones con logaritmos y potencias",
"Logaritmos en base 10",
"Logaritmos",
"Logaritmos naturales y e",
"Logaritmos naturales",
"F�rmula de suma trigonom�trica inversa",
"Forma polar compleja",
"Logaritmos en cualquier base",
"Cambio de base de logaritmos",
"C�lculo de funciones trigonom�tricas",
"Trigonometr�a b�sica",
"Inversas trigonom�tricas",
"Identidad cuadr�tica trigonom�trica",
"Identidad Csc y Cot",
"F�rmula trigonom�trica de suma",
"F�rmula de duplicaciones",
"Desarrollo de cos(nx) y sin(nx)",
"Verificaci�n de identidad",
"Resoluci�n para 30-60-90",
"Resoluci�n para 45-45-90",
"Ceros de funciones trigonom�tricas",
"Funciones trigonom�tricas inversas",
"Simplificaci�n de funciones trigonom�tricas inversas",
"Suma de funciones trigonom�tricas inversas",
"Funciones trigonom�trica complementarias",
"�ngulos complementarios en grados",
"Funciones trigonom�tricas pares y impares",
"Periodicidad de funciones trigonom�tricas",
"F�rmula de bisecci�n",
"Identidad de productos y factores",
"L�mites",
"L�mite de cocientes",
"L�mite de cocientes de ra�ces",
"Regla de L'Hospital",
"L�mites especiales",
"L�mites de funciones hiperb�licas",
"L�mites avanzados",
"L�mites logar�tmicos",
"L�mites y infinit�simos",
"L�mites infinitos",
"Infinitos",
"Denominadores nulos",
"Funciones al infinito",
"Diferenciaci�n polinomial",
"Derivada",
"Diferenciaci�n de funciones trigonom�tricas",
"Diferenciaci�n de exponencial y logaritmos",
"Diferenciaci�n de funciones trigonom�tricas inversas",
"Regla de diferenciaci�n compuesta",
"M�nimos y m�ximos",
"Diferenciaci�n impl�cita",
"Cociente incremental",
"Simplificaci�n",
"Derivada de orden alto",
"Integraci�n de base",
"Integraci�n de funciones trigonom�tricas",
"Integraci�n de funciones trigonom�tricas de ct",
"Integraci�n de exponencial y logaritmos",
"Integraci�n por sustituci�n",
"Integraci�n por partes",
"Teorema fundamental",
"Integral definida",
"Integrales impropias",
"Integrandos pares e impares",
"Sustituci�n inversa",
"Integrales trigonom�tricas",
"Simplificaci�n de integrandos trigonom�tricos",
"Integraci�n de funciones racionales",
"Integraci�n de ra�ces cuadradas en el denominador",
"Integraci�n de funciones trigonom�tricas inversas",
"Simplificar",
"Integraci�n de funciones hiperb�lica",
"Serie geom�trica",
"Serie geom�trica 2",
"Serie geom�trica 3",
"Serie geom�trica 4",
"Serie geom�trica 5",
"Serie infinita para el logaritmo",
"Serie infinita para sin y cos",
"Serie infinita para funciones exponencial",
"Serie infinita para arcotangente",
"Serie infinita para tan y cot",
"Aparici�n de serie",
"Operaciones algebraicas sobre series",
"Manipulaciones de series infinitas",
"Test de convergencia",
"Fin de test de convergencia",
"Funciones complejas",
"Identidad de funciones complejas",
"Seno y coseno iperbolici",
"Identidad trigonom�trica hiperb�lica",
"Funciones hiperb�licos",
"Funciones hiperb�licas inversas",
"Diferenciaci�n hiperb�lica",
"Diferenciaci�n hiperb�lica inversa",
"Funci�n Sg",
"Simplificaci�n de funci�n Sg",
"Funciones de Bessel",
"Funciones de Bessel modificadas",
"Funciones definidas por usarios",
"Invisible", /* automode_only operators */
"Tambi�n invisible", /* Automode_only2 */
"y tamni�n esto" /* Automode_only3 */
};
/*_____________________________________________________________*/
const char **Spanish_cmdmenu(int i)
/* returns an array of strings for the i-th menu */
{ return (const char **) Spanish_menutext[i];
}
const char *Spanish_menutitle(int i)
{ return (const char *) Spanish_menutitles[i];
}
Sindbad File Manager Version 1.0, Coded By Sindbad EG ~ The Terrorists