Anexo:Fórmulas de reducción para integrales

Anexo:Fórmulas de reducción para integrales

En ocasiones la integración definida o indefinida de funciones de una variable se facilita mediante las llamadas fórmulas de reducción. Son éstas una cierta forma de poner en relación integrales que, además de depender de una determinada variable independiente u, también son dependientes de un parámetro n, con otras de la misma (o parecida) especie en las que ese parámetro aparece reducido a otro menor, esto es, fórmulas como


\int f(u,n) \cdot du = g(u,n) + a \int f(u,n-b) \cdot du


Otras veces los parámetros pueden ser más de uno.

La siguiente es una lista de esta clase de fórmulas de reducción, la mayor parte de las veces deducidas mediante la técnica de integración por partes. Cada una de ellas tiene la limitación de no ser aplicable para los respectivos valores de los coeficientes que anulen alguno de los denominadores.


Contenido

Fórmulas de reducción para integrales racionales e irracionales

Que contienen expresiones lineales

I_n = \int \left( \frac {u-a}{u-b} \right)^n du = - \frac {(u-a)^n}{(n-1)(u-b)^{n-1}} + \frac {n}{n-1} I_{n-1}


I_{m,n} = \int \frac {(u-a)^m}{(u-b)^n} \cdot du = - \frac {(u-a)^m}{(n-1)(u-b)^{n-1}} + \frac {m}{n-1} I_{m-1,n-1}


I_{m,n} = \int (u-a)^m (u-b)^n du = \frac {1}{m+n+1}(u-a)^{m+1} (u-b)^n + \frac {n(a-b)}{m+n+1} I_{m,n-1}


I_{m,n} = \int \frac {du}{(u-a)^m (u-b)^n} = \frac {1}{(n-1)(a-b)(x-a)^{m-1} (x-b)^{n-1}} + \frac {m+n+2}{(n-1)(a-b)}I_{m,n-1}


I_{m,n} = \int \frac {u^n du}{(a+bu)^m} = - \frac {u^n}{b(m-1)(a+bu)^{m-1}} + \frac {n}{b(m-1)} I_{m-1,n-1}


I_n = \int \frac {u^n du}{\sqrt {a+bu}} = \frac {2}{b(2n+1)} u^n \sqrt{a+bu} - \frac {2na}{(2n+1)b} I_{n-1}


I_{m,n} = \int \frac {(r+su)^n}{(a+bu)^m} \cdot du = - \frac {(r+su)^n}{(m-1)b(a+bu)^{m-1}} + \frac {ns}{(m-1)b} I_{m-1,n-1}


I_{m,n} = \int \frac {1}{u^n} (a+bu)^m du = - \frac {(a+bu)^m}{(n-1)u^{n-1}} + \frac {mb}{n-1} I_{m-1,n-1}


I_n = \int \frac {1}{u^n} \sqrt {a+bu} \cdot du = - \frac {1}{(n-1)a u^{n-1}} (a+bu)^{3/2} - \frac {(2n-5)b}{2(n-1)a} I_{n-1}


I_{m,n} = \int u^n (a+bu)^m du = \frac {1}{(m+n+1)b} u^n (a+bu)^{m+1} - \frac {na}{(m+n+1)b} I_{m,n-1}


I_{m,n} = \int u^n (a+bu)^m du = \frac {1}{m+n+1} u^{n+1} (a+bu)^m + \frac {ma}{m+n+1} I_{m-1,n}


I_n = \int u^n \sqrt {a+bu} \cdot du = \frac {2}{(2n+3)b} u^n (a+bu)^{3/2} - \frac {2na}{(2n+3)b} I_{n-1}


I_{m,n} = \int (r+su)^n (a+bu)^m du = \frac {1}{(m+n+1)b} (r+su)^n (a+bu)^{m+1} - \frac {n(as-br)}{(m+n+1)b} I_{m,n-1}


I_{m,n} = \int \frac {du}{u^n (a+bu)^m} = - \frac {1}{(n-1) u^{n-1} (a+bu)^m} - \frac {mb}{n-1} I_{m+1,n-1}


I_n = \int \frac {du}{u^n \sqrt {a+bu}} = - \frac {1}{(n-1)a u^{n-1}} \sqrt {a+bu} - \frac {(2n-3)b}{2(n-1)a} I_{n-1}


I_{m,n} = \int \frac {du}{u^n (a+bu)^m} = - \frac {1}{(m-1)b u^n (a+bu)^{m-1}} - \frac {n}{(m-1)b} I_{m-1,n+1}


I_{m,n} = \int \frac {du}{u^n (a+bu)^m} = - \frac {1}{(n-1)a u^{n-1} (a+bu)^{m-1}} - \frac {(m+n-2)b}{(n-1)a} I_{m,n-1}


I_{m,n} = \int \frac {du}{u^n (a+bu)^m} = \frac {1}{(m-1)a u^{n-1} (a+bu)^{m-1}} + \frac {m+n-2}{(m-1)a} I_{m-1,n}


I_{m,n} = \int \frac {du}{(r+su)^n (a+bu)^m} = -\frac {1}{(n-1)(as-br)(r+su)^{n-1} (a+bu)^{m-1}} - \frac {(m+n-2)b}{(n-1)(as-br)} I_{m,n-1}

Que contienen expresiones cuadráticas

I_n = \int \frac {du}{\left( a^2 \pm u^2 \right)^n} = \frac {u}{2a^2 (n-1) \left(a^2 \pm u^2 \right)^{n-1}} + \frac {2n-3}{2a^2(n-1)} I_{n-1}


I_n = \int \frac {du}{\left (u^2 \pm a^2 \right)^n} = \pm \frac {u}{2a^2(n-1) \left( u^2 \pm a^2 \right)^{n-1}}\pm \frac {2n-3}{2a^2(n-1)} I_{n-1}


I_n = \int \left( a^2 \pm u^2 \right)^n du = \frac {u \left( a^2 \pm u^2 \right)^n}{2n+1} + \frac {2a^2 n}{2n+1} I_{n-1}


I_n = \int \left( u^2 - a^2 \right)^n du = \frac {u \left( u^2 - a^2 \right)^n}{2n+1} - \frac {2a^2n}{2n+1} I_{n-2}


I_{m,n} = \int \frac {u^m du}{\left( a^2 \pm u^2 \right)^n} = \mp \frac {u^{m-1}}{2(n-1) \left(a^2 \pm u^2 \right)^{n-1}}\pm \frac {m-1}{2(n-1)} I_{m-2,n-1}


I_n = \int \frac {u^n du}{\sqrt {a^2 - u^2}} = - \frac 1n u^{n-1} \sqrt {a^2 - u^2} + \frac {n-1}{n} a^2 I_{n-2}


I_{m,n} = \int \frac {u^m du}{\left( u^2 \pm a^2 \right)^n} = - \frac {u^{m-1}}{2(n-1) \left(u^2 \pm a^2 \right)^{n-1}} + \frac {m-1}{2(n-1)} I_{m-2,n-1}


I_n = \int \frac {u^n du}{\sqrt{u^2 \pm a^2}} = \frac 1n u^{n-1} \sqrt {u^2 \pm a^2} \mp \frac {n-1}{n} I_{n-2}


I_{m,n} = \int \frac {1}{u^n} \left( a^2 \pm u^2 \right)^m du = -\frac {1}{(n-1) u^{n-1}} \left( a^2 \pm u^2 \right)^m \pm \frac {2m}{n-1} I_{m-1,n-2}


I_{m,n} = \int \frac {1}{u^n} \left( u^2 \pm a^2 \right)^m du = -\frac {1}{(n-1) u^{n-1}} \left( u^2 \pm a^2 \right)^m + \frac {2m}{n-1} I_{m-1,n-2}


I_{m,n} = \int \frac {1}{u^n} \left( a^2 \pm u^2 \right)^m du = \frac {1}{(2m-n+1) u^{n-1}} \left( a^2 \pm u^2 \right)^m + \frac {(n-1)a^2}{2m-n+1} I_{m-1,n}


I_{m,n} = \int \frac {1}{u^n} \left( u^2 \pm a^2 \right)^m du = \frac {1}{(2m-n+1) u^{n-1}} \left( u^2 \pm a^2 \right)^m \pm \frac {(n-1)a^2}{2m-n+1} I_{m-1,n}


I_n = \int \frac {1}{u^n} \sqrt {u^2 \pm a^2} \cdot du = \mp \frac {1}{(n-1) a^2 u^{n-1}} \left( u^2 \pm a^2 \right)^{3/2} \mp \frac {n-4}{(n-1)a^2} I_{n-2}


I_{m,n} = \int \frac {1}{u^n} \left( a^2 \pm u^2 \right)^m du = - \frac {2(m+1) \left( a^2 \pm u^2 \right)^{m+1}}{(n-1)^2 a^2 u^{n-1}} \pm \frac {2(m+1)(2m-n+3)}{(n-1)^2 a^2} I_{m,n-2}


I_n = \int \frac {1}{u^n} \sqrt {a^2 - u^2} \cdot du = \frac {1}{(n-1) a^2 u^{n-1}} \left( a^2 - u^2 \right)^{3/2} - \frac {n-4}{(n-1)a^2} I_{n-2}


I_{m,n} = \int u^n \left( u^2 \pm a^2 \right)^m du = \frac {1}{2m+n+1} u^{n-1} \left( u^2 \pm a^2 \right)^{m+1} \mp \frac {(n-1) a^2}{2m+n+1} I_{m,n-2}


I_n = \int u^n \sqrt {u^2 \pm a^2} \cdot du = \frac {1}{n+2} u^{n-1} \left( u^2 \pm a^2 \right)^{3/2} \mp \frac {n-1}{n+2} a^2 I_{n-2}


I_{m,n} = \int u^n \left( a^2 \pm u^2 \right)^m du = \pm \frac {1}{2m+n+1} u^{n-1} \left( a^2 \pm u^2 \right)^{m+1} \mp \frac {(n-1) a^2}{2m+n+1} I_{m,n-2}


I_n = \int u^n \sqrt {a^2 - u^2} \cdot du = \frac {1}{n+2} u^{n-1} \left( a^2 - u^2 \right)^{3/2} - \frac {n-1}{n+1} a^2 I_{n-2}


I_{m,n} = \int u^n \left( u^2 \pm a^2 \right)^m du = \frac {1}{2m+n+1} u^{n+1} \left( a^2 \pm u^2 \right)^m \pm \frac {2m a^2}{2m+n+1} I_{m-1,n}


I_{m,n} = \int u^n \left( a^2 \pm u^2 \right)^m du = \frac {1}{2m+n+1} u^{n+1} \left( u^2 \pm a^2 \right)^m + \frac {2m a^2}{2m+n+1} I_{m-1,n}


I_{m,n} = \int \frac {du}{u^n \left( a^2 \pm u^2 \right)^m} = \frac {1}{2(m-1)a^2 u^{n-1} \left( a^2 \pm u^2 \right)^{m-1}} + \frac {2m+n-3}{2(m-1)a^2} I_{m-1,n}


I_n = \int \frac {du}{u^n \sqrt {a^2 - u^2}} = - \frac {1}{(n-1)a^2u^{n-1}} \sqrt {a^2 - u^2} + \frac {n-2}{(n-1)a^2} I_{n-2}


I_{m,n} = \int \frac {du}{u^n \left( u^2 \pm a^2 \right)^m} = \pm \frac {1}{2(m-1)a^2 u^{n-1} \left( u^2 \pm a^2 \right)^{m-1}} \pm \frac {2m+n-3}{2(m-1)a^2} I_{m-1,n}


I_n = \int \frac {du}{u^n \sqrt {u^2 \pm a^2}} = \mp \frac {1}{(n-1)a^2 u^{n-1}} \sqrt {u^2 \pm a^2} \mp \frac {n-2}{(n-1)a^2} I_{n-2}


I_{m,n} = \int \frac {du}{u^n \left( a^2 \pm u^2 \right)^m} = - \frac {1}{(n-1)a^2 u^{n-1} \left( a^2 \pm u^2 \right)^{m-1}} \mp \frac {2m+n-3}{(n-1)a^2} I_{m,n-2}


I_{m,n} = \int \frac {du}{u^n \left( u^2 \pm a^2 \right)^m} = \mp \frac {1}{(n-1)a^2 u^{n-1} \left( u^2 \pm a^2 \right)^{m-1}} \mp \frac {2m+n-3}{(n-1)a^2} I_{m,n-2}


I_n = \int \frac {dx}{\left( ax^2+bx+c \right)^n} = \frac {2ax+b}{(n-1) \left( 4ac-b^2 \right) \left( ax^2+bx+c \right)^{n-1}} + \frac {2(2n-3)a}{(n-1) \left( 4ac-b^2 \right)} I_{n-1}

Que contienen otras expresiones

I_m = \int \frac {du}{\left( u^n \pm a^n \right)^m} = \pm \frac {u}{n(m-1)a^n \left( u^n \pm a^n \right)^{m-1}} \pm \frac {n(m-1)-1}{n(m-1)a^n} I_{m-1}


I_m = \int \frac {du}{\left( a^n \pm u^n \right)^m} = \frac {u}{n(m-1)a^n \left( a^n \pm u^n \right)^{m-1}} + \frac {n(m-1)-1}{n(m-1)a^n} I_{m-1}


I_{m,n} = \int \frac {u^m du}{a^n \pm u^n} = \frac {1}{m-n+1} u^{m-n+1} \mp a^n I_{m-n,n}


I_{m,n} = \int \frac {du}{u^m \left( u^n \pm a^n \right)} = \mp \frac {1}{(m-1) u^{m-1}} \mp I_{m-n,n}


I_m = \int \frac {du}{u \left( u^n \pm a^n \right)^m} = \pm \frac {1}{n(m-1) a^n \left(u^n \pm a^n \right)^{m-1}} \pm \frac {1}{a^n} I_{m-1}


I_{r,m} = \int \frac {du}{u^r \left( u^n \pm a^n \right)^m} = \pm \frac {1}{n(m-1)a^n u^{r-1} \left( u^n \pm a^n \right)^{m-1}} \pm \frac {1}{a^n} \left( 1 + \frac {r-1}{n(m-1)} \right) I_{r,m-1}


I_{r,m} = \int \frac {du}{u^r \left( a^n \pm u^n \right)^m} = \frac {1}{n(m-1)a^n u^{r-1} \left( a^n \pm u^n \right)^{m-1}} + \frac {1}{a^n} \left( 1 + \frac {r-1}{n(m-1)} \right) I_{r,m-1}


I_{r,m} = \int \frac {du}{u^r \left( u^n \pm a^n \right)^m} = \mp \frac {1}{(r-1)a^n u^{r-1} \left( u^n \pm a^n \right)^{m-1}} \mp \frac {1}{a^n} \left( 1 + n \frac {m-1}{r-1} \right) I_{r-n,m}


I_{r,m} = \int \frac {du}{u^r \left( a^n \pm u^n \right)^m} = - \frac {1}{(r-1)a^n u^{r-1} \left( a^n \pm u^n \right)^{m-1}} \mp \frac {1}{a^n} \left( 1 + n \frac {m-1}{r-1} \right) I_{r-n,m}


I_{r,m} = \int \frac {u^r du}{\left( u^n \pm a^n \right)^m} = - \frac {u^{r-n+1}}{n(m-1) \left( u^n \pm a^n \right)^{m-1}} \mp \frac {r-n+1}{n(m-1)} I_{r-n,m-1}


I_{r,m} = \int \frac {u^r du}{\left( a^n \pm u^n \right)^m} = \mp \frac {u^{r-n+1}}{n(m-1) \left( a^n \pm u^n \right)^{m-1}} \pm \frac {r-n+1}{n(m-1)} I_{r-n,m-1}


I_{r,m} = \int \frac {u^r du}{\left( u^n \pm a^n \right)^m} = \pm \frac {u^{r+1}}{n(m-1) a^n \left( u^n \pm a^n \right)^{m-1}} \pm \frac {1}{a^n} \left( 1- \frac {r+1}{n(m-1)} \right) I_{r,m-1}


I_{r,m} = \int \frac {u^r du}{\left( a^n \pm u^n \right)^m} = \frac {u^{r+1}}{n(m-1) a^n \left( a^n \pm u^n \right)^{m-1}} + \frac {1}{a^n} \left( 1- \frac {r+1}{n(m-1)} \right) I_{r,m-1}


I_{r,m} = \int \frac {1}{u^r} \left( u^n \pm a^n \right)^m du = - \frac {1}{(r-1) x^{r-1}} \left( u^n \pm a^n \right)^m + \frac {mn}{r-1} I_{r-n,m-1}


I_{r,m} = \int \frac {1}{u^r} \left( a^n \pm u^n \right)^m du = - \frac {1}{(r-1) x^{r-1}} \left( a^n \pm u^n \right)^m \pm \frac {mn}{r-1} I_{r-n,m-1}


I_{r,m} = \int u^r \left( u^n \pm a^n \right)^m du = \frac {1}{mn+r+1} x^{r+1} \left( u^n \pm a^n \right)^m \pm \frac {mn a^n}{mn+r+1} I_{r,m-1}


I_{r,m} = \int u^r \left( a^n \pm u^n \right)^m du = \frac {1}{mn+r+1} x^{r+1} \left( a^n \pm x^n \right)^m + \frac {mn a^n}{mn+r+1} I_{r,m-1}


I_{r,m} = \int u^r \left( u^n \pm a^n \right)^m du = \frac {1}{mn+r+1} x^{r-n+1} \left( u^n \pm a^n \right)^{m+1} \mp \frac {r-n+1}{mn+r+1} a^n I_{r-n,m}


I_{r,m} = \int u^r \left( a^n \pm u^n \right)^m du = \pm \frac {1}{mn+r+1} x^{r-n+1} \left( a^n \pm x^n \right)^{m+1} \mp \frac {r-n+1}{mn+r+1} a^n I_{r-n,m}

Fórmulas de reducción para integrales trigonométricas

Directas

I_n = \int \sin^n u \cdot du = - \frac 1n \sin^{n-1} u \cdot \cos u + \frac {n-1}{n} I_{n-2}


I_n = \int \cos^n u \cdot du = \frac 1n \cos^{n-1} u \cdot \sin u + \frac {n-1}{n} I_{n-2}


I_n = \int \sec^n u \cdot du = \frac {1}{n-1} \sec^{n-2} u \cdot \tan u + \frac {n-2}{n-1} I_{n-2} if "n ≠ 1"


I_n = \int \csc^n u \cdot du = - \frac {1}{n-1} \csc^{n-2} u \cdot \cot u + \frac {n-2}{n-1} I_{n-2}


I_n = \int \tan^n u \cdot du = \frac {1}{n-1} \tan^{n-1} u - I_{n-2}


I_n = \int \cot^n u \cdot du = - \frac {1}{n-1} \cot^{n-1} u - I_{n-2}


I_{m,n} = \int \sin^n u \cdot \cos^n u \cdot du = \frac {1}{m+n} \sin^{m+1} u \cdot \cos^{n-1} u + \frac {n-1}{m+n} I_{m,n-2}


I_{m,n} = \int \sin^n u \cdot \cos^n u \cdot du = - \frac {1}{m+n} \sin^{m-1} u \cdot \cos^{n+1} u + \frac {m-1}{m+n} I_{m-2,n}


I_{m,n} = \int \frac {\sin^m u}{\cos^n u} \cdot du = \frac {\sin^{m-1} u}{(n-1) \cos^{n-1} u} - \frac {m-1}{n-1} I_{m-2,n-2}


I_{m,n} = \int \frac {\sin^m u}{\cos^n u} \cdot du = \frac {\sin^{m+1} u}{(n-1) \cos^{n-1} u} - \frac {m-n+2}{n-1} I_{m,n-2}


I_{m,n} = \int \frac {\sin^m u}{\cos^n u} \cdot du = - \frac {\sin^{m-1} u}{(m-n) \cos^{n-1} u} + \frac {m-1}{m-2} I_{m-2,n}


I_{m,n} = \int \frac {\cos^m u}{\sin^n u} \cdot du = - \frac {\cos^{m-1} u}{(n-1) \sin^{n-1} u} - \frac {m-1}{n-1} I_{m-2,n-2}


I_{m,n} = \int \frac {\cos^m u}{\sin^n u} \cdot du = - \frac {\cos^{m+1} u}{(n-1) \sin^{n-1} u} - \frac {m-n+2}{n-1} I_{m,n-2}


I_{m,n} = \int \frac {\cos^m u}{\sin^n u} \cdot du = \frac {\cos^{m-1} u}{(m-n) \sin^{n-1} u} + \frac {m-1}{m-2} I_{m-2,n}


I_n = \int \frac {\sin^n u}{\cos u} \cdot du = - \frac {1}{(n-1)} \sin^{n-1} u + I_{n-2}


I_n = \int \frac {\cos^n u}{\sin u} \cdot du = \frac {1}{(n-1)} \cos^{n-1} u + I_{n-2}


I_{m,n} = \int \frac {du}{\sin^m u \cdot \cos^n u} = \frac {1}{(n-1) \sin^{m-1} u \cdot \cos^{n-1} u} + \frac {m+n-2}{n-1} I_{m,n-2}


I_{m,n} = \int \frac {du}{\sin^m u \cdot \cos^n u} = - \frac {1}{(m-1) \sin^{m-1} u \cdot \cos^{n-1} u} + \frac {m+n-2}{m-1} I_{m-2,n}


I_n = \int \frac {du}{\sin^n u \cdot \cos u} = - \frac {1}{(n-1) \sin^{n-1} u } + I_{n-2}


I_n = \int \frac {du}{\sin u \cdot \cos^n u} = \frac {1}{(n-1) \cos^{n-1} u } + I_{n-2}


I_n = \int \cos nu \cdot \cos^n u \cdot du = \frac {1}{2n} \sin nu \cdot \cos^n u + \frac 12 I_{n-1}


I_n = \int \sin nu \cdot cos^n u \cdot du = - \frac {1}{2n} \cos nu \cdot \cos^n u + \frac 12 I_{n-1}


I_n = \int \cos nu \cdot sin^n u \cdot du = \frac {1}{2n} \sin nu \cdot \sin^n u - \frac 12 \int \sin (n-1)u \cdot \sin^{n-1} u \cdot du


I_n = \int \sin nu \cdot sin^n u \cdot du = - \frac {1}{2n} \cos nu \cdot \sin^n u + \frac 12 \int \cos (n-1)u \cdot \sin^{n-1} u \cdot du


I_{m,n} = \int \cos mu \cdot \cos^n u \cdot du = \frac {1}{m+n} \sin mu \cdot \cos^n u + \frac {n}{m+n} I_{m-1,n-1}


I_{m,n} = \int \sin mu \cdot \cos^n u \cdot du = - \frac {1}{m+n} \cos mu \cdot \cos^n u + \frac {n}{m+n} I_{m-1,n-1}


I_{m,n} = \int \cos mu \cdot \sin^n u \cdot du = \frac {1}{m+n} \sin mu \cdot \sin^n u - \frac {n}{m+n} \int \sin (m-1)u \cdot \sin^{n-1} u \cdot du


I_{m,n} = \int \sin mu \cdot \sin^n u \cdot du = - \frac {1}{m+n} \cos mu \cdot \sin^n u + \frac {n}{m+n} \int \cos (m-1)u \cdot \sin^{n-1} u \cdot du


I_n = \int \frac {\cos nu}{\cos^n u} \cdot du = - \frac {\sin (n-1)u}{(n-1) \cos^{n-1} u} + 2 I_{n-1}


I_n = \int \frac {\sin nu}{\cos^n u} \cdot du = \frac {\cos (n-1)u}{(n-1) \cos^{n-1} u} + 2 I_{n-1}


I_n = \int \frac {\sin nu}{\sin^n u} \cdot du = - \frac {\sin (n-1)u}{(n-1) \sin^{n-1} u} + 2 \int \frac {\cos (n-1)u}{\sin^{n-1}u} \cdot du


I_n = \int \frac {\cos nu}{\sin^n u} \cdot du = - \frac {\cos (n-1)u}{(n-1) \sin^{n-1} u} - 2 \int \frac {\sin (n-1)u}{\sin^{n-1}u} \cdot du


I_{m,n} = \int \frac {\cos mu}{\cos^n u} \cdot du = - \frac {\sin (m-1)u}{(n-1) \cos^{n-1} u} + \frac {m+n-2}{n-1} I_{m-1,n-1}


I_{m,n} = \int \frac {\sin mu}{\cos^n u} \cdot du = \frac {\cos (m-1)u}{(n-1) \cos^{n-1} u} + \frac {m+n-2}{n-1} I_{m-1,n-1}


I_{m,n} = \int \frac {\sin mu}{\sin^n u} \cdot du = - \frac {\sin (m-1)u}{(n-1) \sin^{n-1} u} + \frac {m+n-2}{n-1} \int \frac {\cos (m-1)u}{\sin^{n-1} u} \cdot du


I_{m,n} = \int \frac {\cos mu}{\sin^n u} \cdot du = - \frac {\cos (m-1)u}{(n-1) \sin^{n-1} u} - \frac {m+n-2}{n-1} \int \frac {\sin (m-1)u}{\sin^{n-1} u} \cdot du


I_n = \int (\sin u \pm \cos u)^n du = - \frac 1n (\cos u \mp \sin u)(\sin u \pm \cos u)^{n-1} + 2 \frac {n-1}{n} I_{n-2}


I_n = \int (\cos u \pm \sin u)^n du = \frac 1n (\sin u \mp \cos u)(\cos u \pm \sin u)^{n-1} + 2 \frac {n-1}{n} I_{n-2}


I_n = \int \frac {du}{(\sin u \pm \cos u)^n} = - \frac {\cos u \mp \sin u}{2(n-1)(\sin u \pm \cos u)^{n-1}} + \frac {n-2}{2(n-1)} I_{n-2}


I_n = \int \frac {du}{(\cos u \pm \sin u)^n} = \frac {\sin u \mp \cos u}{2(n-1)(\cos u \pm \sin u)^{n-1}} + \frac {n-2}{2(n-1)} I_{n-2}


I_n = \int (a \sin u + b \cos u)^n du = - \frac 1n (a \cos u - b \sin u)(a \sin u + b \cos u)^{n-1} + \frac {n-1}{n} (a^2 + b^2) I_{n-2}


I_n = \int \frac {du}{(a \sin u + b \cos u)^n} = - \frac {a \cos u - b \sin u}{(n-1)(a^2 + b^2)(a \sin u + b \cos u)^{n-1}} + \frac {n-2}{(n-1)(a^2 + b^2)} I_{n-2}


I_n = \int u^n \sin au \cdot du = - \frac 1a u^n \cos au + \frac na \int u^{n-1} \cos au \cdot du


I_n = \int u^n \cos au \cdot du = \frac 1a u^n \sin au - \frac na \int u^{n-1} \sin au \cdot du


I_n = \int \frac {1}{u^n} \sin au \cdot du = - \frac {1}{(n-1) u^{n-1}} \sin au + \frac {a}{n-1} \int \frac {1}{u^{n-1}} \cos au \cdot du


I_n = \int \frac {1}{u^n} \cos au \cdot du = - \frac {1}{(n-1) u^{n-1}} \cos au - \frac {a}{n-1} \int \frac {1}{u^{n-1}} \sin au \cdot du


I_n = \int u \sin^n au \cdot du = \frac {1}{a^2 n^2} (\sin au - nau \cos au) \sin^{n-1} au + \frac {n-1}{n} I_{n-2}


I_n = \int u \cos^n au \cdot du = \frac {1}{a^2 n^2} (\cos au - nau \sin au) \cos^{n-1} au + \frac {n-1}{n} I_{n-2}


I_n = \int u \sec^n au \cdot du = \frac {u}{(n-1)a} \sec^{n-2} au \cdot \tan au - \frac {1}{(n-1)(n-2)a^2} \sec^{n-2} au + \frac {n-2}{n-1} I_{n-2}


I_n = \int u \csc^n au \cdot du = - \frac {u}{(n-1)a} \csc^{n-2} au \cdot \cot au - \frac {1}{(n-1)(n-2)a^2} \csc^{n-2} au + \frac {n-2}{n-1} I_{n-2}

Inversas

I_n = \int (\arcsin u)^n du = \left( u \arcsin u + n \sqrt {1-u^2} \right)(\arcsin u)^{n-1} - n(n-1) I_{n-2}


I_n = \int (\arccos u)^n du = \left( u \arccos u - n \sqrt {1-u^2} \right)(\arccos u)^{n-1} - n(n-1) I_{n-2}


I_n = \int \frac {du}{(\arcsin u)^n} = \frac {u \arcsin u - (n-2) \sqrt{1-x^2}}{(n-1)(n-2)(\arcsin u)^{n-1}} - \frac 1 {(n-1)(n-2)}I_{n-2}


I_n = \int \frac {du}{(\arccos u)^n} = \frac {u \arccos u + (n-2) \sqrt{1-x^2}}{(n-1)(n-2)(\arccos u)^{n-1}} - \frac 1 {(n-1)(n-2)}I_{n-2}


I_n = \int u^n \arcsin u \cdot du = \frac {1}{n+1} u^{n+1} \arcsin u - \frac{1}{n+1} \int \frac {u^{n+1} du}{\sqrt {1-u^2}}


I_n = \int u^n \arccos u \cdot du = \frac 1{n+1} u^{n+1} \arccos u + \frac 1{n+1} \int \frac {u^{n+1}du}{\sqrt {1-u^2}}


I_n = \int u^n \arcsin u \cdot du = \frac {1}{n+1} u^n \left( u \arcsin u + \sqrt {1-u^2} \right) - \frac {n}{n+1} \int u^{n-1} \sqrt {1-u^2} \cdot du


I_n = \int u^n \arccos u \cdot du = \frac 1{n+1} u^n \left( u \arccos u - \sqrt {1-u^2} \right) + \frac n{n+1} \int u^{n-1} \sqrt {1-u^2} \cdot du


I_n = \int \frac {1}{u^n} \arcsin u \cdot du = - \frac {1}{(n-1) u^{n-1}} \arcsin u + \frac {1}{n-1} \int \frac {du}{u^{n-1} \sqrt {1-u^2}}


I_n = \int \frac 1{u^n} \arccos u \cdot du = - \frac 1{(n-1) u^{n-1}} \arccos u - \frac 1{n-1} \int \frac {du}{u^{n-1} \sqrt {1-u^2}}


I_n = \int u^n \arctan u \cdot du = \frac {1}{n+1} u^{n+1} \arctan u - \frac {1}{n+1} \int \frac {u^{n+1} du}{1+u^2}


I_n = \int u^n \arccot u \cdot du2 = \frac 1{n+1} u^{n+1} \arccot u + \frac 1{n+1} \int \frac {u^{n+1 }du}{1 + u^2}


I_n = \int \frac 1{u^n} \arctan u \cdot du = - \frac 1{(n-1) u^{n-1}} \arctan u + \frac 1{n-1} \int \frac {du}{u^{n-1} (1+u^2)}


I_n = \int \frac 1{u^n} \arccot u \cdot du = - \frac 1{(n-1) u^{n-1}} \arccot u - \frac 1{n-1} \int \frac {du}{u^{n-1} (1+u^2)}

Fórmulas de reducción para integrales exponenciales

I_n = \int u^n e^{au} du = \frac 1a u^n e^{au} - \frac na I_{n-1}


I_n = \int \frac{1}{u^n} e^{au} du = -\frac {1}{(n-1) u^{n-1}} e^{au} - \frac {a}{n-1} I_{n-1}


I_n = \int u^n e^{a u^2} du = \frac {1}{2a} u^{n-1} e^{a u^2} - \frac {n-1}{2a} I_{n-2}


I_n = \int e^{au} \sin^n bu \cdot du = \frac {1}{a^2 + b^2n^2} e^{au} (a \sin bu - nb \cos bu) \sin^{n-1} bu + \frac {n(n-1)b^2}{a^2 + b^2n^2} I_{n-2}


I_n = \int e^{au} \cos^n bu \cdot du = \frac {1}{a^2 + b^2n^2} e^{au} (a \cos bu - nb \sin bu) \cos^{n-1} bu + \frac {n(n-1)b^2}{a^2 + b^2n^2} I_{n-2}


{I_n} = \int {{x^n}{e^{ - x}}} dx = - {e^{ - x}}\left[ {\sum\limits_{k = 0}^n {{{{d^k}\left( {{x^n}} \right)} \over {d{x^k}}}} } \right];{{{d^0}\left( {{x^n}} \right)} \over {d{x^0}}} = {x^n};{{{d^n}\left( {{x^n}} \right)} \over {d{x^n}}} = n!

Fórmulas de reducción para integrales logarítmicas

I_n = \int \ln^n u \cdot du = u \ln^n u - n I_{n-1}


I_n = \int \frac {du}{\ln^n u} = - \frac {u}{(n-1) \ln^{n-1} u} + \frac {1}{n-1} I_{n-1}


I_n = \int u^m \ln^n u \cdot du = \frac {1}{m+1} u^{m+1} \ln^n u - \frac {n}{m+1} I_{n-1}


I_n = \int \frac {1}{u^m} \ln^n u \cdot du = - \frac {1}{(m-1) u^{m-1}} \ln ^n u + \frac {n}{m-1} I_{n-1}


I_n = \int \frac {u^m}{\ln^n u} \cdot du = - \frac {u^{m+1}}{(n-1) \ln^{n-1} u} + \frac {m+1}{n-1} I_{n-1}


I_n = \int \frac {du}{u^m \ln^n u} = - \frac {1}{(n-1) u^{m-1} \ln^{n-1} u} - \frac {m-1}{n-1} I_{n-1}

Fórmulas de reducción para integrales hiperbólicas

I_n = \int \sinh^n u \cdot du = \frac 1n \sinh^{n-1} u \cdot \cosh u - \frac {n-1}{n} I_{n-2}


I_n = \int \cosh^n u \cdot du = \frac 1n \cosh^{n-1} u \cdot \sinh u + \frac {n-1}{n} I_{n-2}


I_n = \int \mathrm {sech^n u} \cdot du = \frac {1}{n-1} \mathrm {sech^{n-2} u} \cdot \tanh u + \frac {n-2}{n-1} I_{n-2}


I_n = \int \mathrm {csch^n u} \cdot du = - \frac {1}{n-1} \mathrm {csch^{n-2} u} \cdot \coth u - \frac {n-2}{n-1} I_{n-2}


I_n = \int \tanh^n u \cdot du = - \frac 1{n-1} \tanh^{n-1} u + I_{n-2}


I_n = \int \coth^n u \cdot du = - \frac 1{n-1} \coth^{n-1} u + I_{n-2}


I_{m,n} = \int \sinh^m u \cdot \cosh^n u \cdot du = \frac {1}{m+n} \sinh^{m-1} u \cdot \cosh^{n+1} u - \frac {m-1}{m+n} I_{m-2,n}


I_{m,n} = \int \sinh^m u \cdot \cosh^n u \cdot du = \frac {1}{m+n} \sinh^{m+1} u \cdot \cosh^{n-1} u + \frac {n-1}{m+n} I_{m,n-2}


I_{m,n} = \int \frac {\sinh^m u}{\cosh^n u} \cdot du = - \frac {\sinh^{m-1} u}{(n-1) \cosh^{n-1} u} + \frac {m-1}{n-1} I_{m-2,n-2}


I_{m,n} = \int \frac {\cosh^m u}{\sinh^n u} \cdot du = - \frac {\cosh^{m-1} u}{(n-1) \sinh^{n-1} u} + \frac {m-1}{n-1} I_{m-2,n-2}


I_n = \int u^n \sinh au \cdot du = \frac 1a u^n \cosh au - \frac na \int u^{n-1} \cosh au \cdot du


I_n = \int u^n \cosh au \cdot du = \frac 1a u^n \sinh au - \frac na \int u^{n-1} \sinh au \cdot du

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