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 ${\displaystyle u}$, también son dependientes de un parámetro ${\displaystyle n}$, con otras de la misma (o parecida) especie en las que ese parámetro aparece reducido a otro menor, esto es, fórmulas como

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle I_{n}=\int {\frac {du}{\left(au^{2}+bu+c\right)^{n))}={\frac {2au+b}{(n-1)\left(4ac-b^{2}\right)\left(au^{2}+bu+c\right)^{n-1))}+{\frac {2(2n-3)a}{(n-1)\left(4ac-b^{2}\right)))I_{n-1))$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle I_{n}=\int \operatorname {sen} ^{n}u\,du=-{\frac {1}{n))\operatorname {sen} ^{n-1}u\,\cos u+{\frac {n-1}{n))I_{n-2))$

${\displaystyle I_{n}=\int \cos ^{n}u\,du={\frac {1}{n))\cos ^{n-1}u\,\operatorname {sen} u+{\frac {n-1}{n))I_{n-2))$

${\displaystyle I_{n}=\int \sec ^{n}u\,du={\frac {1}{n-1))\sec ^{n-2}u\,\tan u+{\frac {n-2}{n-1))I_{n-2))$

${\displaystyle I_{n}=\int \csc ^{n}u\,du=-{\frac {1}{n-1))\csc ^{n-2}u\,\cot u+{\frac {n-2}{n-1))I_{n-2))$

${\displaystyle I_{n}=\int \tan ^{n}u\,du={\frac {1}{n-1))\tan ^{n-1}u-I_{n-2))$

${\displaystyle I_{n}=\int \cot ^{n}u\,du=-{\frac {1}{n-1))\cot ^{n-1}u-I_{n-2))$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle I_{n}=\int \cos nu\,\cos ^{n}u\,du={\frac {1}{2n))\operatorname {sen} nu\,\cos ^{n}u+{\frac {1}{2))I_{n-1))$

${\displaystyle I_{n}=\int \operatorname {sen} nu\,\cos ^{n}u\,du=-{\frac {1}{2n))\cos nu\,\cos ^{n}u+{\frac {1}{2))I_{n-1))$