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.

Fórmulas de reducción para integrales racionales e irracionales

Que contienen expresiones lineales

$I_n=\int {\left(\frac{au\pm b}{pu\pm q}\right)}^{n}du=-\frac{(au\pm b{)}^n}{p(n-1)(pu\pm q{)}^{n-1}}+\frac{an}{p(n-1)}{I}_{n-1}$

$I_n=\int {\left(\frac{b\pm au}{q\pm pu}\right)}^{n}du=\mp \frac{(b\pm au{)}^n}{p(n-1)(q\pm pu{)}^{n-1}}+\frac{an}{p(n-1)}{I}_{n-1}$

$I_{m,n}=\int \frac{du}{(au\pm b{)}^m(pu\pm q{)}^n}=\frac{1}{a(n-1)(\mp bp\pm aq)(au\pm b{)}^{m-1}(pu\pm q{)}^{n-1}}+\frac{m+n-2}{(n-1)(\mp bp\pm aq)}{I}_{m,n-1}$

$I_{m,n}=\int {(au\pm b)}^m{(pu\pm q)}^n=\frac{{(au\pm b)}^{m+1}{(pu\pm q)}^n}{a(m+n+1)}-\frac{n}{a}\frac{(\mp aq\pm bp)}{(m+n+1)}{I}_{m,n-1}$

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

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

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

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

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

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

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

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

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

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

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

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

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

Que contienen expresiones cuadráticas

$I_n=\int \frac{du}{{(a^2\pm u^2)}^n}=\frac{u}{2a^2(n-1){(a^2\pm u^2)}^{n-1}}+\frac{2n-3}{2a^2(n-1)}{I}_{n-1}$

$I_n=\int \frac{du}{{(u^2\pm a^2)}^n}=\pm \frac{u}{2a^2(n-1){(u^2\pm a^2)}^{n-1}}\pm \frac{2n-3}{2a^2(n-1)}{I}_{n-1}$

$I_n=\int {(a^2\pm u^2)}^{n}du=\frac{u{(a^2\pm u^2)}^n}{2n+1}+\frac{2a^{2}n}{2n+1}{I}_{n-1}$

$I_n=\int {(u^2-a^2)}^{n}du=\frac{u{(u^2-a^2)}^n}{2n+1}-\frac{2a^{2}n}{2n+1}{I}_{n-1}$

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

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

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

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

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

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

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

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

$I_n=\int \frac{1}{u^n}\sqrt{u^2\pm a^2}du=\mp \frac{1}{(n-1)a^2{u}^{n-1}}{(u^2\pm a^2)}^{3/2}\mp \frac{n-4}{(n-1)a^2}{I}_{n-2}$

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

$I_n=\int \frac{1}{u^n}\sqrt{a^2-u^2}du=\frac{1}{(n-1)a^2{u}^{n-1}}{(a^2-u^2)}^{3/2}-\frac{n-4}{(n-1)a^2}{I}_{n-2}$

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

$I_n=\int u^n\sqrt{u^2\pm a^2}du=\frac{1}{n+2}{u}^{n-1}{(u^2\pm a^2)}^{3/2}\mp \frac{n-1}{n+2}{a}^2{I}_{n-2}$

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

$I_n=\int u^n\sqrt{a^2-u^2}du=\frac{1}{n+2}{u}^{n-1}{(a^2-u^2)}^{3/2}-\frac{n-1}{n+1}{a}^2{I}_{n-2}$

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

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

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

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

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

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

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

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

$I_n=\int \frac{du}{{(a{u}^2+bu+c)}^n}=\frac{2au+b}{(n-1)(4ac-b^2){(a{u}^2+bu+c)}^{n-1}}+\frac{2(2n-3)a}{(n-1)(4ac-b^2)}{I}_{n-1}$

Que contienen otras expresiones

$I_m=\int \frac{du}{{(u^n\pm a^n)}^m}=\pm \frac{u}{n(m-1)a^n{(u\pm a^n)}^{m-1}}\pm \frac{n(m-1)-1}{n(m-1)a^n}{I}_{m-1}$

$I_m=\int \frac{du}{{(a^n\pm u^n)}^m}=\frac{u}{n(m-1)a^n{(a^n\pm u^n)}^{m-1}}+\frac{n(m-1)-1}{n(m-1)a^n}{I}_{m-1}$

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

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

$I_m=\int \frac{du}{u{(u^n\pm a^n)}^m}=\pm \frac{1}{n(m-1)a^n{(u^n\pm a^n)}^{m-1}}\pm \frac{1}{a^n}{I}_{m-1}$

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

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

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

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

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

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

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

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

$I_{r,m}=\int \frac{1}{u^r}{(u^n\pm a^n)}^{m}du=-\frac{1}{(r-1)u^{r-1}}{(u^n\pm a^n)}^m+\frac{mn}{r-1}{I}_{r-n,m-1}$

$I_{r,m}=\int \frac{1}{u^r}{(a^n\pm u^n)}^{m}du=-\frac{1}{(r-1)u^{r-1}}{(a^n\pm u^n)}^m\pm \frac{mn}{r-1}{I}_{r-n,m-1}$

$I_{r,m}=\int u^r{(u^n\pm a^n)}^{m}du=\frac{1}{mn+r+1}{u}^{r+1}{(u^n\pm a^n)}^m\pm \frac{mn{a}^n}{mn+r+1}{I}_{r,m-1}$

$I_{r,m}=\int u^r{(a^n\pm u^n)}^{m}du=\frac{1}{mn+r+1}{u}^{r+1}{(a^n\pm u^n)}^m+\frac{mn{a}^n}{mn+r+1}{I}_{r,m-1}$

$I_{r,m}=\int u^r{(u^n\pm a^n)}^{m}du=\frac{1}{mn+r+1}{u}^{r-n+1}{(u^n\pm a^n)}^{m+1}\mp \frac{r-n+1}{mn+r+1}{a}^n{I}_{r-n,m}$

$I_{r,m}=\int u^r{(a^n\pm u^n)}^{m}du=\pm \frac{1}{mn+r+1}{u}^{r-n+1}{(a^n\pm u^n)}^{m+1}\mp \frac{r-n+1}{mn+r+1}{a}^n{I}_{r-n,m}$

Fórmulas de reducción para integrales trigonométricas

Directas

$I_n=\int {\mathrm{sen}}^{n}udu=-\frac{1}{n}{\mathrm{sen}}^{n-1}u\cos u+\frac{n-1}{n}{I}_{n-2}$

$I_n=\int {\cos }^{n}udu=\frac{1}{n}{\cos }^{n-1}u\mathrm{sen}u+\frac{n-1}{n}{I}_{n-2}$

$I_n=\int {\sec }^{n}udu=\frac{1}{n-1}{\sec }^{n-2}u\tan u+\frac{n-2}{n-1}{I}_{n-2}$

$I_n=\int {\csc }^{n}udu=-\frac{1}{n-1}{\csc }^{n-2}u\cot u+\frac{n-2}{n-1}{I}_{n-2}$

$I_n=\int {\tan }^{n}udu=\frac{1}{n-1}{\tan }^{n-1}u-I_{n-2}$

$I_n=\int {\cot }^{n}udu=-\frac{1}{n-1}{\cot }^{n-1}u-I_{n-2}$

$I_{m,n}=\int {\mathrm{sen}}^{m}u{\cos }^{n}udu=\frac{1}{m+n}{\mathrm{sen}}^{m+1}u{\cos }^{n-1}u+\frac{n-1}{m+n}{I}_{m,n-2}$

$I_{m,n}=\int {\mathrm{sen}}^{m}u{\cos }^{n}udu=-\frac{1}{m+n}{\mathrm{sen}}^{m-1}u{\cos }^{n+1}u+\frac{m-1}{m+n}{I}_{m-2,n}$

$I_{m,n}=\int \frac{{\mathrm{sen}}^{m}u}{{\cos }^{n}u}du=\frac{{\mathrm{sen}}^{m-1}u}{(n-1){\cos }^{n-1}u}-\frac{m-1}{n-1}{I}_{m-2,n-2}$

$I_{m,n}=\int \frac{{\mathrm{sen}}^{m}u}{{\cos }^{n}u}du=\frac{{\mathrm{sen}}^{m+1}u}{(n-1){\cos }^{n-1}u}-\frac{m-n+2}{n-1}{I}_{m,n-2}$

$I_{m,n}=\int \frac{{\mathrm{sen}}^{m}u}{{\cos }^{n}u}du=-\frac{{\mathrm{sen}}^{m-1}u}{(m-n){\cos }^{n-1}u}+\frac{m-1}{m-2}{I}_{m-2,n}$

$I_{m,n}=\int \frac{{\cos }^{m}u}{{\mathrm{sen}}^{n}u}du=-\frac{{\cos }^{m-1}u}{(n-1){\mathrm{sen}}^{n-1}u}-\frac{m-1}{n-1}{I}_{m-2,n-2}$

$I_{m,n}=\int \frac{{\cos }^{m}u}{{\mathrm{sen}}^{n}u}du=-\frac{{\cos }^{m+1}u}{(n-1){\mathrm{sen}}^{n-1}u}-\frac{m-n+2}{n-1}{I}_{m,n-2}$

$I_{m,n}=\int \frac{{\cos }^{m}u}{{\mathrm{sen}}^{n}u}du=\frac{{\cos }^{m-1}u}{(m-n){\mathrm{sen}}^{n-1}u}+\frac{m-1}{m-2}{I}_{m-2,n}$

$I_n=\int \frac{{\mathrm{sen}}^{n}u}{\cos u}du=-\frac{1}{(n-1)}{\mathrm{sen}}^{n-1}u+I_{n-2}$

$I_n=\int \frac{{\cos }^{n}u}{\mathrm{sen}u}du=\frac{1}{(n-1)}{\cos }^{n-1}u+I_{n-2}$

$I_{m,n}=\int \frac{du}{{\mathrm{sen}}^{m}u{\cos }^{n}u}=\frac{1}{(n-1){\mathrm{sen}}^{m-1}u{\cos }^{n-1}u}+\frac{m+n-2}{n-1}{I}_{m,n-2}$

$I_{m,n}=\int \frac{du}{{\mathrm{sen}}^{m}u{\cos }^{n}u}=-\frac{1}{(m-1){\mathrm{sen}}^{m-1}u{\cos }^{n-1}u}+\frac{m+n-2}{m-1}{I}_{m-2,n}$

$I_n=\int \frac{du}{{\mathrm{sen}}^{n}u\cos u}=-\frac{1}{(n-1){\mathrm{sen}}^{n-1}u}+I_{n-2}$

$I_n=\int \frac{du}{\mathrm{sen}u{\cos }^{n}u}=\frac{1}{(n-1){\cos }^{n-1}u}+I_{n-2}$

$I_n=\int \cos nu{\cos }^{n}udu=\frac{1}{2n}\mathrm{sen}nu{\cos }^{n}u+\frac{1}{2}{I}_{n-1}$

$I_n=\int \mathrm{sen}nu{\cos }^{n}udu=-\frac{1}{2n}\cos nu{\cos }^{n}u+\frac{1}{2}{I}_{n-1}$

$I_n=\int \cos nu{\mathrm{sen}}^{n}udu=\frac{1}{2n}\mathrm{sen}nu{\mathrm{sen}}^{n}u-\frac{1}{2}\int \mathrm{sen}(n-1)u{\mathrm{sen}}^{n-1}udu$

$I_n=\int \mathrm{sen}nu{\mathrm{sen}}^{n}udu=-\frac{1}{2n}\cos nu{\mathrm{sen}}^{n}u+\frac{1}{2}\int \cos (n-1)u{\mathrm{sen}}^{n-1}udu$

$I_{m,n}=\int \cos mu{\cos }^{n}udu=\frac{1}{m+n}\mathrm{sen}mu{\cos }^{n}u+\frac{n}{m+n}{I}_{m-1,n-1}$

$I_{m,n}=\int \mathrm{sen}mu{\cos }^{n}udu=-\frac{1}{m+n}\cos mu{\cos }^{n}u+\frac{n}{m+n}{I}_{m-1,n-1}$

$I_{m,n}=\int \cos mu{\mathrm{sen}}^{n}udu=\frac{1}{m+n}\mathrm{sen}mu{\mathrm{sen}}^{n}u-\frac{n}{m+n}\int \mathrm{sen}(m-1)u{\mathrm{sen}}^{n-1}udu$

$I_{m,n}=\int \mathrm{sen}mu{\mathrm{sen}}^{n}udu=-\frac{1}{m+n}\cos mu{\mathrm{sen}}^{n}u+\frac{n}{m+n}\int \cos (m-1)u{\mathrm{sen}}^{n-1}udu$

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

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

$I_n=\int \frac{\mathrm{sen}nu}{{\mathrm{sen}}^{n}u}du=-\frac{\mathrm{sen}(n-1)u}{(n-1){\mathrm{sen}}^{n-1}u}+2\int \frac{\cos (n-1)u}{{\mathrm{sen}}^{n-1}u}du$

$I_n=\int \frac{\cos nu}{{\mathrm{sen}}^{n}u}du=-\frac{\cos (n-1)u}{(n-1){\mathrm{sen}}^{n-1}u}-2\int \frac{\mathrm{sen}(n-1)u}{{\mathrm{sen}}^{n-1}u}du$

$I_{m,n}=\int \frac{\cos mu}{{\cos }^{n}u}du=-\frac{\mathrm{sen}(m-1)u}{(n-1){\cos }^{n-1}u}+\frac{m+n-2}{n-1}{I}_{m-1,n-1}$

$I_{m,n}=\int \frac{\mathrm{sen}mu}{{\cos }^{n}u}du=\frac{\cos (m-1)u}{(n-1){\cos }^{n-1}u}+\frac{m+n-2}{n-1}{I}_{m-1,n-1}$

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

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

$I_n=\int (\mathrm{sen}u\pm \cos u{)}^{n}du=-\frac{1}{n}(\cos u\mp \mathrm{sen}u)(\mathrm{sen}u\pm \cos u{)}^{n-1}+2\frac{n-1}{n}{I}_{n-2}$

$I_n=\int (\cos u\pm \mathrm{sen}u{)}^{n}du=\frac{1}{n}(\mathrm{sen}u\mp \cos u)(\cos u\pm \mathrm{sen}u{)}^{n-1}+2\frac{n-1}{n}{I}_{n-2}$

$I_n=\int \frac{du}{(\mathrm{sen}u\pm \cos u{)}^n}=-\frac{\cos u\mp \mathrm{sen}u}{2(n-1)(\mathrm{sen}u\pm \cos u{)}^{n-1}}+\frac{n-2}{2(n-1)}{I}_{n-2}$

$I_n=\int \frac{du}{(\cos u\pm \mathrm{sen}u{)}^n}=\frac{\mathrm{sen}u\mp \cos u}{2(n-1)(\cos u\pm \mathrm{sen}u{)}^{n-1}}+\frac{n-2}{2(n-1)}{I}_{n-2}$

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

$I_n=\int \frac{du}{(a\mathrm{sen}u+b\cos u{)}^n}=-\frac{a\cos u-b\mathrm{sen}u}{(n-1)(a^2+b^2)(a\mathrm{sen}u+b\cos u{)}^{n-1}}+\frac{n-2}{(n-1)(a^2+b^2)}{I}_{n-2}$

$I_n=\int u^n\mathrm{sen}audu=-\frac{1}{a}{u}^n\cos au+\frac{n}{a}\int u^{n-1}\cos audu$

$I_n=\int u^n\cos audu=\frac{1}{a}{u}^n\mathrm{sen}au-\frac{n}{a}\int u^{n-1}\mathrm{sen}audu$

$I_n=\int \frac{1}{u^n}\mathrm{sen}audu=-\frac{1}{(n-1)u^{n-1}}\mathrm{sen}au+\frac{a}{n-1}\int \frac{1}{u^{n-1}}\cos audu$

$I_n=\int \frac{1}{u^n}\cos audu=-\frac{1}{(n-1)u^{n-1}}\cos au-\frac{a}{n-1}\int \frac{1}{u^{n-1}}\mathrm{sen}audu$

$I_n=\int u{\mathrm{sen}}^{n}audu=\frac{1}{a^2{n}^2}(\mathrm{sen}au-nau\cos au){\mathrm{sen}}^{n-1}au+\frac{n-1}{n}{I}_{n-2}$

$I_n=\int u{\cos }^{n}audu=\frac{1}{a^2{n}^2}(\cos au-nau\mathrm{sen}au){\cos }^{n-1}au+\frac{n-1}{n}{I}_{n-2}$

$I_n=\int u{\sec }^{n}audu=\frac{u}{(n-1)a}{\sec }^{n-2}au\tan au-\frac{1}{(n-1)(n-2)a^2}{\sec }^{n-2}au+\frac{n-2}{n-1}{I}_{n-2}$

$I_n=\int u{\csc }^{n}audu=-\frac{u}{(n-1)a}{\csc }^{n-2}au\cot au-\frac{1}{(n-1)(n-2)a^2}{\csc }^{n-2}au+\frac{n-2}{n-1}{I}_{n-2}$

Inversas

$I_n=\int {\arcsin }^{n}udu=(u\arcsin u+n\sqrt{1-u^2})(\arcsin u{)}^{n-1}-n(n-1)I_{n-2}$

$I_n=\int {\arccos }^{n}udu=(u\arccos u-n\sqrt{1-u^2})(\arccos u{)}^{n-1}-n(n-1)I_{n-2}$

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

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

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

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

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

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

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

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

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

$I_n=\int u^n\mathrm{arccot}udu=\frac{1}{n+1}{u}^{n+1}\mathrm{arccot}u+\frac{1}{n+1}\int \frac{u^{n+1}du}{1+u^2}$

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

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

Fórmulas de reducción para integrales exponenciales

$I_n=\int u^n{e}^{au}du=\frac{1}{a}{u}^n{e}^{au}-\frac{n}{a}{I}_{n-1}$

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

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

$I_n=\int e^{au}{\mathrm{sen}}^{n}budu=\frac{1}{a^2+b^2{n}^2}{e}^{au}(a\mathrm{sen}bu-nb\cos bu){\mathrm{sen}}^{n-1}bu+\frac{n(n-1)b^2}{a^2+b^2{n}^2}{I}_{n-2}$

$I_n=\int e^{au}{\cos }^{n}budu=\frac{1}{a^2+b^2{n}^2}{e}^{au}(a\cos bu-nb\mathrm{sen}bu){\cos }^{n-1}bu+\frac{n(n-1)b^2}{a^2+b^2{n}^2}{I}_{n-2}$

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

Fórmulas de reducción para integrales logarítmicas

$I_n=\int {\ln }^{n}udu=u{\ln }^{n}u-n{I}_{n-1}$

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

$I_n=\int u^m{\ln }^{n}udu=\frac{1}{m+1}{u}^{m+1}{\ln }^{n}u-\frac{n}{m+1}{I}_{n-1}$

$I_n=\int \frac{1}{u^m}{\ln }^{n}udu=-\frac{1}{(m-1)u^{m-1}}{\ln }^{n}u+\frac{n}{m-1}{I}_{n-1}$

$I_n=\int \frac{u^m}{{\ln }^{n}u}du=-\frac{u^{m+1}}{(n-1){\ln }^{n-1}u}+\frac{m+1}{n-1}{I}_{n-1}$

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

Fórmulas de reducción para integrales hiperbólicas

$I_n=\int {\sinh }^{n}udu=\frac{1}{n}{\sinh }^{n-1}u\cosh u-\frac{n-1}{n}{I}_{n-2}$

$I_n=\int {\cosh }^{n}udu=\frac{1}{n}{\cosh }^{n-1}u\sinh u+\frac{n-1}{n}{I}_{n-2}$

$I_n=\int \mathrm{s}\mathrm{e}\mathrm{c}{\mathrm{h}}^{\mathrm{n}}\mathrm{u}du=\frac{1}{n-1}\mathrm{s}\mathrm{e}\mathrm{c}{\mathrm{h}}^{\mathrm{n}\mathrm{-}\mathrm{2}}\mathrm{u}\tanh u+\frac{n-2}{n-1}{I}_{n-2}$

$I_n=\int \mathrm{c}\mathrm{s}\mathrm{c}{\mathrm{h}}^{\mathrm{n}}\mathrm{u}du=-\frac{1}{n-1}\mathrm{c}\mathrm{s}\mathrm{c}{\mathrm{h}}^{\mathrm{n}\mathrm{-}\mathrm{2}}\mathrm{u}\coth u-\frac{n-2}{n-1}{I}_{n-2}$

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

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

$I_{m,n}=\int {\sinh }^{m}u{\cosh }^{n}udu=\frac{1}{m+n}{\sinh }^{m-1}u{\cosh }^{n+1}u-\frac{m-1}{m+n}{I}_{m-2,n}$

$I_{m,n}=\int {\sinh }^{m}u{\cosh }^{n}udu=\frac{1}{m+n}{\sinh }^{m+1}u{\cosh }^{n-1}u+\frac{n-1}{m+n}{I}_{m,n-2}$

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

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

$I_n=\int u^n\sinh audu=\frac{1}{a}{u}^n\cosh au-\frac{n}{a}\int u^{n-1}\cosh audu$

$I_n=\int u^n\cosh audu=\frac{1}{a}{u}^n\sinh au-\frac{n}{a}\int u^{n-1}\sinh audu$