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
{\displaystyle u}
n
{\displaystyle n}
∫
f
(
u
,
n
)
⋅
d
u
=
g
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u
,
n
)
+
a
∫
f
(
u
,
n
−
b
)
⋅
d
u
{\displaystyle \int f(u,n)\cdot du=g(u,n)+a\int f(u,n-b)\cdot du}
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.
Que contienen expresiones lineales
editar
I
n
=
∫
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a
u
±
b
p
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±
q
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n
d
u
=
−
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b
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n
p
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n
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1
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n
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a
n
p
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n
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I
n
−
1
{\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}}
I
n
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b
±
a
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q
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p
u
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u
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p
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u
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a
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p
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n
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I
n
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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}}
I
m
,
n
=
∫
d
u
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a
u
±
b
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m
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a
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q
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1
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2
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n
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1
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p
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a
q
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I
m
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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}}
I
m
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n
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a
u
±
b
)
m
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=
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m
+
1
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a
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−
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a
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b
p
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m
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1
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I
m
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n
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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}}
I
m
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=
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n
d
u
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a
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u
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m
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u
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m
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n
b
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m
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1
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I
m
−
1
,
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}}
I
n
=
∫
u
n
d
u
a
+
b
u
=
2
b
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2
n
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1
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u
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u
−
2
n
a
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2
n
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1
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b
I
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}}
I
m
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n
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∫
(
r
+
s
u
)
n
(
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u
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m
d
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n
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m
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s
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m
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b
I
m
−
1
,
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}}
I
m
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n
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1
u
n
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a
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b
u
)
m
d
u
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m
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1
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u
n
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1
+
m
b
n
−
1
I
m
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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}}
I
n
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∫
1
u
n
a
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d
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1
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n
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1
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1
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a
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u
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3
/
2
−
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2
n
−
5
)
b
2
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n
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)
a
I
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}}
I
m
,
n
=
∫
u
n
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a
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b
u
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m
d
u
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1
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1
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b
u
n
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u
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m
+
1
−
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a
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m
+
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+
1
)
b
I
m
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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}}
I
m
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u
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d
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m
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1
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a
+
b
u
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m
+
m
a
m
+
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+
1
I
m
−
1
,
n
{\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}}
I
n
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∫
u
n
a
+
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d
u
=
2
(
2
n
+
3
)
b
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n
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a
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b
u
)
3
/
2
−
2
n
a
(
2
n
+
3
)
b
I
n
−
1
{\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}}
I
m
,
n
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u
u
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u
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m
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1
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n
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1
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u
n
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1
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a
+
b
u
)
m
−
m
b
n
−
1
I
m
+
1
,
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}}
I
n
=
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d
u
u
n
a
+
b
u
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1
(
n
−
1
)
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u
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1
a
+
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u
−
(
2
n
−
3
)
b
2
(
n
−
1
)
a
I
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}}
I
m
,
n
=
∫
d
u
u
n
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a
+
b
u
)
m
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−
1
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m
−
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b
u
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u
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m
−
1
−
n
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m
−
1
)
b
I
m
−
1
,
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}}
I
m
,
n
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∫
d
u
u
n
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a
+
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u
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m
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1
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n
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1
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a
u
n
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1
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a
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u
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m
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1
−
(
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}{(n-1)au^{n-1}(a+bu)^{m-1}}}-{\frac {(m+n-2)b}{(n-1)a}}I_{m,n-1}}
I
m
,
n
=
∫
d
u
u
n
(
a
+
b
u
)
m
=
1
(
m
−
1
)
a
u
n
−
1
(
a
+
b
u
)
m
−
1
+
m
+
n
−
2
(
m
−
1
)
a
I
m
−
1
,
n
{\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}}
editar
I
n
=
∫
d
u
(
a
2
±
u
2
)
n
=
u
2
a
2
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n
−
1
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(
a
2
±
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2
)
n
−
1
+
2
n
−
3
2
a
2
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n
−
1
)
I
n
−
1
{\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}}
I
n
=
∫
d
u
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u
2
±
a
2
)
n
=
±
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2
a
2
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n
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1
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2
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n
−
1
±
2
n
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3
2
a
2
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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}}
I
n
=
∫
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a
2
±
u
2
)
n
d
u
=
u
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a
2
±
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2
)
n
2
n
+
1
+
2
a
2
n
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}}
I
n
=
∫
(
u
2
−
a
2
)
n
d
u
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u
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2
−
a
2
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n
2
n
+
1
−
2
a
2
n
2
n
+
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}}
I
m
,
n
=
∫
u
m
d
u
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a
2
±
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2
)
n
=
∓
u
m
−
1
2
(
n
−
1
)
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a
2
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2
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n
−
1
±
m
−
1
2
(
n
−
1
)
I
m
−
2
,
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}}
I
n
=
∫
u
n
d
u
a
2
−
u
2
=
−
1
n
u
n
−
1
a
2
−
u
2
+
n
−
1
n
a
2
I
n
−
2
{\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}}
I
m
,
n
=
∫
u
m
d
u
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u
2
±
a
2
)
n
=
−
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m
−
1
2
(
n
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1
)
(
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2
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a
2
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n
−
1
+
m
−
1
2
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n
−
1
)
I
m
−
2
,
n
−
1
{\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}}
I
n
=
∫
u
n
d
u
u
2
±
a
2
=
1
n
u
n
−
1
u
2
±
a
2
∓
n
−
1
n
I
n
−
2
{\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}}
I
m
,
n
=
∫
1
u
n
(
a
2
±
u
2
)
m
d
u
=
−
1
(
n
−
1
)
u
n
−
1
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a
2
±
u
2
)
m
±
2
m
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}{(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
=
∫
1
u
n
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u
2
±
a
2
)
m
d
u
=
−
1
(
n
−
1
)
u
n
−
1
(
u
2
±
a
2
)
m
+
2
m
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}}
I
m
,
n
=
∫
1
u
n
(
a
2
±
u
2
)
m
d
u
=
1
(
2
m
−
n
+
1
)
u
n
−
1
(
a
2
±
u
2
)
m
+
(
n
−
1
)
a
2
2
m
−
n
+
1
I
m
−
1
,
n
{\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}}
I
m
,
n
=
∫
1
u
n
(
u
2
±
a
2
)
m
d
u
=
1
(
2
m
−
n
+
1
)
u
n
−
1
(
u
2
±
a
2
)
m
±
(
n
−
1
)
a
2
2
m
−
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}}
I
n
=
∫
1
u
n
u
2
±
a
2
d
u
=
∓
1
(
n
−
1
)
a
2
u
n
−
1
(
u
2
±
a
2
)
3
/
2
∓
n
−
4
(
n
−
1
)
a
2
I
n
−
2
{\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}}
I
m
,
n
=
∫
1
u
n
(
a
2
±
u
2
)
m
d
u
=
−
2
(
m
+
1
)
(
a
2
±
u
2
)
m
+
1
(
n
−
1
)
2
a
2
u
n
−
1
±
2
(
m
+
1
)
(
2
m
−
n
+
3
)
(
n
−
1
)
2
a
2
I
m
,
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}}
I
n
=
∫
1
u
n
a
2
−
u
2
d
u
=
1
(
n
−
1
)
a
2
u
n
−
1
(
a
2
−
u
2
)
3
/
2
−
n
−
4
(
n
−
1
)
a
2
I
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}}
I
m
,
n
=
∫
u
n
(
u
2
±
a
2
)
m
d
u
=
1
2
m
+
n
+
1
u
n
−
1
(
u
2
±
a
2
)
m
+
1
∓
(
n
−
1
)
a
2
2
m
+
n
+
1
I
m
,
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}}
I
n
=
∫
u
n
u
2
±
a
2
d
u
=
1
n
+
2
u
n
−
1
(
u
2
±
a
2
)
3
/
2
∓
n
−
1
n
+
2
a
2
I
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}}
I
m
,
n
=
∫
u
n
(
a
2
±
u
2
)
m
d
u
=
±
1
2
m
+
n
+
1
u
n
−
1
(
a
2
±
u
2
)
m
+
1
∓
(
n
−
1
)
a
2
2
m
+
n
+
1
I
m
,
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}}
I
n
=
∫
u
n
a
2
−
u
2
d
u
=
1
n
+
2
u
n
−
1
(
a
2
−
u
2
)
3
/
2
−
n
−
1
n
+
1
a
2
I
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}}
I
m
,
n
=
∫
u
n
(
u
2
±
a
2
)
m
d
u
=
1
2
m
+
n
+
1
u
n
+
1
(
a
2
±
u
2
)
m
±
2
m
a
2
2
m
+
n
+
1
I
m
−
1
,
n
{\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}}
I
m
,
n
=
∫
u
n
(
a
2
±
u
2
)
m
d
u
=
1
2
m
+
n
+
1
u
n
+
1
(
u
2
±
a
2
)
m
+
2
m
a
2
2
m
+
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}}
I
m
,
n
=
∫
d
u
u
n
(
a
2
±
u
2
)
m
=
1
2
(
m
−
1
)
a
2
u
n
−
1
(
a
2
±
u
2
)
m
−
1
+
2
m
+
n
−
3
2
(
m
−
1
)
a
2
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}}
I
n
=
∫
d
u
u
n
a
2
−
u
2
=
−
1
(
n
−
1
)
a
2
u
n
−
1
a
2
−
u
2
+
n
−
2
(
n
−
1
)
a
2
I
n
−
2
{\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}}
I
m
,
n
=
∫
d
u
u
n
(
u
2
±
a
2
)
m
=
±
1
2
(
m
−
1
)
a
2
u
n
−
1
(
u
2
±
a
2
)
m
−
1
±
2
m
+
n
−
3
2
(
m
−
1
)
a
2
I
m
−
1
,
n
{\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}}
I
n
=
∫
d
u
u
n
u
2
±
a
2
=
∓
1
(
n
−
1
)
a
2
u
n
−
1
u
2
±
a
2
∓
n
−
2
(
n
−
1
)
a
2
I
n
−
2
{\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}}
I
m
,
n
=
∫
d
u
u
n
(
a
2
±
u
2
)
m
=
−
1
(
n
−
1
)
a
2
u
n
−
1
(
a
2
±
u
2
)
m
−
1
∓
2
m
+
n
−
3
(
n
−
1
)
a
2
I
m
,
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}}
I
m
,
n
=
∫
d
u
u
n
(
u
2
±
a
2
)
m
=
∓
1
(
n
−
1
)
a
2
u
n
−
1
(
u
2
±
a
2
)
m
−
1
∓
2
m
+
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}}
I
n
=
∫
d
u
(
a
u
2
+
b
u
+
c
)
n
=
2
a
u
+
b
(
n
−
1
)
(
4
a
c
−
b
2
)
(
a
u
2
+
b
u
+
c
)
n
−
1
+
2
(
2
n
−
3
)
a
(
n
−
1
)
(
4
a
c
−
b
2
)
I
n
−
1
{\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}}
Que contienen otras expresiones
editar
I
m
=
∫
d
u
(
u
n
±
a
n
)
m
=
±
u
n
(
m
−
1
)
a
n
(
u
±
a
n
)
m
−
1
±
n
(
m
−
1
)
−
1
n
(
m
−
1
)
a
n
I
m
−
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}}
I
m
=
∫
d
u
(
a
n
±
u
n
)
m
=
u
n
(
m
−
1
)
a
n
(
a
n
±
u
n
)
m
−
1
+
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}}
I
m
,
n
=
∫
u
m
d
u
a
n
±
u
n
=
1
m
−
n
+
1
u
m
−
n
+
1
∓
a
n
I
m
−
n
,
n
{\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}}
I
m
,
n
=
∫
d
u
u
m
(
u
n
±
a
n
)
=
∓
1
(
m
−
1
)
u
m
−
1
∓
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}}
I
m
=
∫
d
u
u
(
u
n
±
a
n
)
m
=
±
1
n
(
m
−
1
)
a
n
(
u
n
±
a
n
)
m
−
1
±
1
a
n
I
m
−
1
{\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}}
I
r
,
m
=
∫
d
u
u
r
(
u
n
±
a
n
)
m
=
±
1
n
(
m
−
1
)
a
n
u
r
−
1
(
u
n
±
a
n
)
m
−
1
±
1
a
n
(
1
+
r
−
1
n
(
m
−
1
)
)
I
r
,
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}}
I
r
,
m
=
∫
d
u
u
r
(
a
n
±
u
n
)
m
=
1
n
(
m
−
1
)
a
n
u
r
−
1
(
a
n
±
u
n
)
m
−
1
+
1
a
n
(
1
+
r
−
1
n
(
m
−
1
)
)
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}}
I
r
,
m
=
∫
d
u
u
r
(
u
n
±
a
n
)
m
=
∓
1
(
r
−
1
)
a
n
u
r
−
1
(
u
n
±
a
n
)
m
−
1
∓
1
a
n
(
1
+
n
m
−
1
r
−
1
)
I
r
−
n
,
m
{\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}}
I
r
,
m
=
∫
d
u
u
r
(
a
n
±
u
n
)
m
=
−
1
(
r
−
1
)
a
n
u
r
−
1
(
a
n
±
u
n
)
m
−
1
∓
1
a
n
(
1
+
n
m
−
1
r
−
1
)
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}}
I
r
,
m
=
∫
u
r
d
u
(
u
n
±
a
n
)
m
=
−
u
r
−
n
+
1
n
(
m
−
1
)
(
u
n
±
a
n
)
m
−
1
∓
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}}}=-{\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
=
∫
u
r
d
u
(
a
n
±
u
n
)
m
=
∓
u
r
−
n
+
1
n
(
m
−
1
)
(
a
n
±
u
n
)
m
−
1
±
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}}
I
r
,
m
=
∫
u
r
d
u
(
u
n
±
a
n
)
m
=
±
u
r
+
1
n
(
m
−
1
)
a
n
(
u
n
±
a
n
)
m
−
1
±
1
a
n
(
1
−
r
+
1
n
(
m
−
1
)
)
I
r
,
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}}
I
r
,
m
=
∫
u
r
d
u
(
a
n
±
u
n
)
m
=
u
r
+
1
n
(
m
−
1
)
a
n
(
a
n
±
u
n
)
m
−
1
+
1
a
n
(
1
−
r
+
1
n
(
m
−
1
)
)
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}}
I
r
,
m
=
∫
1
u
r
(
u
n
±
a
n
)
m
d
u
=
−
1
(
r
−
1
)
u
r
−
1
(
u
n
±
a
n
)
m
+
m
n
r
−
1
I
r
−
n
,
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}}
I
r
,
m
=
∫
1
u
r
(
a
n
±
u
n
)
m
d
u
=
−
1
(
r
−
1
)
u
r
−
1
(
a
n
±
u
n
)
m
±
m
n
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}}
I
r
,
m
=
∫
u
r
(
u
n
±
a
n
)
m
d
u
=
1
m
n
+
r
+
1
u
r
+
1
(
u
n
±
a
n
)
m
±
m
n
a
n
m
n
+
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+1}\left(u^{n}\pm a^{n}\right)^{m}\pm {\frac {mna^{n}}{mn+r+1}}I_{r,m-1}}
I
r
,
m
=
∫
u
r
(
a
n
±
u
n
)
m
d
u
=
1
m
n
+
r
+
1
u
r
+
1
(
a
n
±
u
n
)
m
+
m
n
a
n
m
n
+
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}}
I
r
,
m
=
∫
u
r
(
u
n
±
a
n
)
m
d
u
=
1
m
n
+
r
+
1
u
r
−
n
+
1
(
u
n
±
a
n
)
m
+
1
∓
r
−
n
+
1
m
n
+
r
+
1
a
n
I
r
−
n
,
m
{\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}}
I
r
,
m
=
∫
u
r
(
a
n
±
u
n
)
m
d
u
=
±
1
m
n
+
r
+
1
u
r
−
n
+
1
(
a
n
±
u
n
)
m
+
1
∓
r
−
n
+
1
m
n
+
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}}
I
n
=
∫
sen
n
u
d
u
=
−
1
n
sen
n
−
1
u
cos
u
+
n
−
1
n
I
n
−
2
{\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}}
I
n
=
∫
cos
n
u
d
u
=
1
n
cos
n
−
1
u
sen
u
+
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}}
I
n
=
∫
sec
n
u
d
u
=
1
n
−
1
sec
n
−
2
u
tan
u
+
n
−
2
n
−
1
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}}
I
n
=
∫
csc
n
u
d
u
=
−
1
n
−
1
csc
n
−
2
u
cot
u
+
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}}
I
n
=
∫
tan
n
u
d
u
=
1
n
−
1
tan
n
−
1
u
−
I
n
−
2
{\displaystyle I_{n}=\int \tan ^{n}u\,du={\frac {1}{n-1}}\tan ^{n-1}u-I_{n-2}}
I
n
=
∫
cot
n
u
d
u
=
−
1
n
−
1
cot
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}}
I
m
,
n
=
∫
sen
m
u
cos
n
u
d
u
=
1
m
+
n
sen
m
+
1
u
cos
n
−
1
u
+
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 {n-1}{m+n}}I_{m,n-2}}
I
m
,
n
=
∫
sen
m
u
cos
n
u
d
u
=
−
1
m
+
n
sen
m
−
1
u
cos
n
+
1
u
+
m
−
1
m
+
n
I
m
−
2
,
n
{\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}}
I
m
,
n
=
∫
sen
m
u
cos
n
u
d
u
=
sen
m
−
1
u
(
n
−
1
)
cos
n
−
1
u
−
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-1}{n-1}}I_{m-2,n-2}}
I
m
,
n
=
∫
sen
m
u
cos
n
u
d
u
=
sen
m
+
1
u
(
n
−
1
)
cos
n
−
1
u
−
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}{(n-1)\cos ^{n-1}u}}-{\frac {m-n+2}{n-1}}I_{m,n-2}}
I
m
,
n
=
∫
sen
m
u
cos
n
u
d
u
=
−
sen
m
−
1
u
(
m
−
n
)
cos
n
−
1
u
+
m
−
1
m
−
2
I
m
−
2
,
n
{\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}}
I
m
,
n
=
∫
cos
m
u
sen
n
u
d
u
=
−
cos
m
−
1
u
(
n
−
1
)
sen
n
−
1
u
−
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-1}{n-1}}I_{m-2,n-2}}
I
m
,
n
=
∫
cos
m
u
sen
n
u
d
u
=
−
cos
m
+
1
u
(
n
−
1
)
sen
n
−
1
u
−
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}{(n-1)\operatorname {sen} ^{n-1}u}}-{\frac {m-n+2}{n-1}}I_{m,n-2}}
I
m
,
n
=
∫
cos
m
u
sen
n
u
d
u
=
cos
m
−
1
u
(
m
−
n
)
sen
n
−
1
u
+
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}{(m-n)\operatorname {sen} ^{n-1}u}}+{\frac {m-1}{m-2}}I_{m-2,n}}
I
n
=
∫
sen
n
u
cos
u
d
u
=
−
1
(
n
−
1
)
sen
n
−
1
u
+
I
n
−
2
{\displaystyle I_{n}=\int {\frac {\operatorname {sen} ^{n}u}{\cos u}}\,du=-{\frac {1}{(n-1)}}\operatorname {sen} ^{n-1}u+I_{n-2}}
I
n
=
∫
cos
n
u
sen
u
d
u
=
1
(
n
−
1
)
cos
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}}
I
m
,
n
=
∫
d
u
sen
m
u
cos
n
u
=
1
(
n
−
1
)
sen
m
−
1
u
cos
n
−
1
u
+
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}{(n-1)\operatorname {sen} ^{m-1}u\,\cos ^{n-1}u}}+{\frac {m+n-2}{n-1}}I_{m,n-2}}
I
m
,
n
=
∫
d
u
sen
m
u
cos
n
u
=
−
1
(
m
−
1
)
sen
m
−
1
u
cos
n
−
1
u
+
m
+
n
−
2
m
−
1
I
m
−
2
,
n
{\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}}
I
n
=
∫
d
u
sen
n
u
cos
u
=
−
1
(
n
−
1
)
sen
n
−
1
u
+
I
n
−
2
{\displaystyle I_{n}=\int {\frac {du}{\operatorname {sen} ^{n}u\,\cos u}}=-{\frac {1}{(n-1)\operatorname {sen} ^{n-1}u}}+I_{n-2}}
I
n
=
∫
d
u
sen
u
cos
n
u
=
1
(
n
−
1
)
cos
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}}
I
n
=
∫
cos
n
u
cos
n
u
d
u
=
1
2
n
sen
n
u
cos
n
u
+
1
2
I
n
−
1
{\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}}
I
n
=
∫
sen
n
u
cos
n
u
d
u
=
−
1
2
n
cos
n
u
cos
n
u
+
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}}
I
n
=
∫
cos
n
u
sen
n
u
d
u
=
1
2
n
sen
n
u
sen
n
u
−
1
2
∫
sen
(
n
−
1
)
u
sen
n
−
1
u
d
u
{\displaystyle I_{n}=\int \cos nu\,\operatorname {sen} ^{n}u\,du={\frac {1}{2n}}\operatorname {sen} nu\,\operatorname {sen} ^{n}u-{\frac {1}{2}}\int \operatorname {sen}(n-1)u\,\operatorname {sen} ^{n-1}u\,du}
I
n
=
∫
sen
n
u
sen
n
u
d
u
=
−
1
2
n
cos
n
u
sen
n
u
+
1
2
∫
cos
(
n
−
1
)
u
sen
n
−
1
u
d
u
{\displaystyle I_{n}=\int \operatorname {sen} nu\,\operatorname {sen} ^{n}u\,du=-{\frac {1}{2n}}\cos nu\,\operatorname {sen} ^{n}u+{\frac {1}{2}}\int \cos(n-1)u\,\operatorname {sen} ^{n-1}u\,du}
I
m
,
n
=
∫
cos
m
u
cos
n
u
d
u
=
1
m
+
n
sen
m
u
cos
n
u
+
n
m
+
n
I
m
−
1
,
n
−
1
{\displaystyle I_{m,n}=\int \cos mu\,\cos ^{n}u\,du={\frac {1}{m+n}}\operatorname {sen} mu\,\cos ^{n}u+{\frac {n}{m+n}}I_{m-1,n-1}}
I
m
,
n
=
∫
sen
m
u
cos
n
u
d
u
=
−
1
m
+
n
cos
m
u
cos
n
u
+
n
m
+
n
I
m
−
1
,
n
−
1
{\displaystyle I_{m,n}=\int \operatorname {sen} mu\,\cos ^{n}u\,du=-{\frac {1}{m+n}}\cos mu\,\cos ^{n}u+{\frac {n}{m+n}}I_{m-1,n-1}}
I
m
,
n
=
∫
cos
m
u
sen
n
u
d
u
=
1
m
+
n
sen
m
u
sen
n
u
−
n
m
+
n
∫
sen
(
m
−
1
)
u
sen
n
−
1
u
d
u
{\displaystyle I_{m,n}=\int \cos mu\,\operatorname {sen} ^{n}u\,du={\frac {1}{m+n}}\operatorname {sen} mu\,\operatorname {sen} ^{n}u-{\frac {n}{m+n}}\int \operatorname {sen}(m-1)u\,\operatorname {sen} ^{n-1}u\,du}
I
m
,
n
=
∫
sen
m
u
sen
n
u
d
u
=
−
1
m
+
n
cos
m
u
sen
n
u
+
n
m
+
n
∫
cos
(
m
−
1
)
u
sen
n
−
1
u
d
u
{\displaystyle I_{m,n}=\int \operatorname {sen} mu\,\operatorname {sen} ^{n}u\,du=-{\frac {1}{m+n}}\cos mu\,\operatorname {sen} ^{n}u+{\frac {n}{m+n}}\int \cos(m-1)u\,\operatorname {sen} ^{n-1}u\,du}
I
n
=
∫
cos
n
u
cos
n
u
d
u
=
−
sen
(
n
−
1
)
u
(
n
−
1
)
cos
n
−
1
u
+
2
I
n
−
1
{\displaystyle I_{n}=\int {\frac {\cos nu}{\cos ^{n}u}}\,du=-{\frac {\operatorname {sen}(n-1)u}{(n-1)\cos ^{n-1}u}}+2I_{n-1}}
I
n
=
∫
sen
n
u
cos
n
u
d
u
=
cos
(
n
−
1
)
u
(
n
−
1
)
cos
n
−
1
u
+
2
I
n
−
1
{\displaystyle I_{n}=\int {\frac {\operatorname {sen} nu}{\cos ^{n}u}}\,du={\frac {\cos(n-1)u}{(n-1)\cos ^{n-1}u}}+2I_{n-1}}
I
n
=
∫
sen
n
u
sen
n
u
d
u
=
−
sen
(
n
−
1
)
u
(
n
−
1
)
sen
n
−
1
u
+
2
∫
cos
(
n
−
1
)
u
sen
n
−
1
u
d
u
{\displaystyle I_{n}=\int {\frac {\operatorname {sen} nu}{\operatorname {sen} ^{n}u}}\,du=-{\frac {\operatorname {sen}(n-1)u}{(n-1)\operatorname {sen} ^{n-1}u}}+2\int {\frac {\cos(n-1)u}{\operatorname {sen} ^{n-1}u}}\,du}
I
n
=
∫
cos
n
u
sen
n
u
d
u
=
−
cos
(
n
−
1
)
u
(
n
−
1
)
sen
n
−
1
u
−
2
∫
sen
(
n
−
1
)
u
sen
n
−
1
u
d
u
{\displaystyle I_{n}=\int {\frac {\cos nu}{\operatorname {sen} ^{n}u}}\,du=-{\frac {\cos(n-1)u}{(n-1)\operatorname {sen} ^{n-1}u}}-2\int {\frac {\operatorname {sen}(n-1)u}{\operatorname {sen} ^{n-1}u}}\,du}
I
m
,
n
=
∫
cos
m
u
cos
n
u
d
u
=
−
sen
(
m
−
1
)
u
(
n
−
1
)
cos
n
−
1
u
+
m
+
n
−
2
n
−
1
I
m
−
1
,
n
−
1
{\displaystyle I_{m,n}=\int {\frac {\cos mu}{\cos ^{n}u}}\,du=-{\frac {\operatorname {sen}(m-1)u}{(n-1)\cos ^{n-1}u}}+{\frac {m+n-2}{n-1}}I_{m-1,n-1}}
I
m
,
n
=
∫
sen
m
u
cos
n
u
d
u
=
cos
(
m
−
1
)
u
(
n
−
1
)
cos
n
−
1
u
+
m
+
n
−
2
n
−
1
I
m
−
1
,
n
−
1
{\displaystyle I_{m,n}=\int {\frac {\operatorname {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
=
∫
sen
m
u
sen
n
u
d
u
=
−
sen
(
m
−
1
)
u
(
n
−
1
)
sen
n
−
1
u
+
m
+
n
−
2
n
−
1
∫
cos
(
m
−
1
)
u
sen
n
−
1
u
d
u
{\displaystyle I_{m,n}=\int {\frac {\operatorname {sen} mu}{\operatorname {sen} ^{n}u}}\,du=-{\frac {\operatorname {sen}(m-1)u}{(n-1)\operatorname {sen} ^{n-1}u}}+{\frac {m+n-2}{n-1}}\int {\frac {\cos(m-1)u}{\operatorname {sen} ^{n-1}u}}\,du}
I
m
,
n
=
∫
cos
m
u
sen
n
u
d
u
=
−
cos
(
m
−
1
)
u
(
n
−
1
)
sen
n
−
1
u
−
m
+
n
−
2
n
−
1
∫
sen
(
m
−
1
)
u
sen
n
−
1
u
d
u
{\displaystyle I_{m,n}=\int {\frac {\cos mu}{\operatorname {sen} ^{n}u}}\,du=-{\frac {\cos(m-1)u}{(n-1)\operatorname {sen} ^{n-1}u}}-{\frac {m+n-2}{n-1}}\int {\frac {\operatorname {sen}(m-1)u}{\operatorname {sen} ^{n-1}u}}\,du}
I
n
=
∫
(
sen
u
±
cos
u
)
n
d
u
=
−
1
n
(
cos
u
∓
sen
u
)
(
sen
u
±
cos
u
)
n
−
1
+
2
n
−
1
n
I
n
−
2
{\displaystyle I_{n}=\int (\operatorname {sen} u\pm \cos u)^{n}du=-{\frac {1}{n}}(\cos u\mp \operatorname {sen} u)(\operatorname {sen} u\pm \cos u)^{n-1}+2{\frac {n-1}{n}}I_{n-2}}
I
n
=
∫
(
cos
u
±
sen
u
)
n
d
u
=
1
n
(
sen
u
∓
cos
u
)
(
cos
u
±
sen
u
)
n
−
1
+
2
n
−
1
n
I
n
−
2
{\displaystyle I_{n}=\int (\cos u\pm \operatorname {sen} u)^{n}du={\frac {1}{n}}(\operatorname {sen} u\mp \cos u)(\cos u\pm \operatorname {sen} u)^{n-1}+2{\frac {n-1}{n}}I_{n-2}}
I
n
=
∫
d
u
(
sen
u
±
cos
u
)
n
=
−
cos
u
∓
sen
u
2
(
n
−
1
)
(
sen
u
±
cos
u
)
n
−
1
+
n
−
2
2
(
n
−
1
)
I
n
−
2
{\displaystyle I_{n}=\int {\frac {du}{(\operatorname {sen} u\pm \cos u)^{n}}}=-{\frac {\cos u\mp \operatorname {sen} u}{2(n-1)(\operatorname {sen} u\pm \cos u)^{n-1}}}+{\frac {n-2}{2(n-1)}}I_{n-2}}
I
n
=
∫
d
u
(
cos
u
±
sen
u
)
n
=
sen
u
∓
cos
u
2
(
n
−
1
)
(
cos
u
±
sen
u
)
n
−
1
+
n
−
2
2
(
n
−
1
)
I
n
−
2
{\displaystyle I_{n}=\int {\frac {du}{(\cos u\pm \operatorname {sen} u)^{n}}}={\frac {\operatorname {sen} u\mp \cos u}{2(n-1)(\cos u\pm \operatorname {sen} u)^{n-1}}}+{\frac {n-2}{2(n-1)}}I_{n-2}}
I
n
=
∫
(
a
sen
u
+
b
cos
u
)
n
d
u
=
−
1
n
(
a
cos
u
−
b
sen
u
)
(
a
sen
u
+
b
cos
u
)
n
−
1
+
n
−
1
n
(
a
2
+
b
2
)
I
n
−
2
{\displaystyle I_{n}=\int (a\operatorname {sen} u+b\cos u)^{n}du=-{\frac {1}{n}}(a\cos u-b\operatorname {sen} u)(a\operatorname {sen} u+b\cos u)^{n-1}+{\frac {n-1}{n}}(a^{2}+b^{2})I_{n-2}}
I
n
=
∫
d
u
(
a
sen
u
+
b
cos
u
)
n
=
−
a
cos
u
−
b
sen
u
(
n
−
1
)
(
a
2
+
b
2
)
(
a
sen
u
+
b
cos
u
)
n
−
1
+
n
−
2
(
n
−
1
)
(
a
2
+
b
2
)
I
n
−
2
{\displaystyle I_{n}=\int {\frac {du}{(a\operatorname {sen} u+b\cos u)^{n}}}=-{\frac {a\cos u-b\operatorname {sen} u}{(n-1)(a^{2}+b^{2})(a\operatorname {sen} u+b\cos u)^{n-1}}}+{\frac {n-2}{(n-1)(a^{2}+b^{2})}}I_{n-2}}
I
n
=
∫
u
n
sen
a
u
d
u
=
−
1
a
u
n
cos
a
u
+
n
a
∫
u
n
−
1
cos
a
u
d
u
{\displaystyle I_{n}=\int u^{n}\operatorname {sen} au\,du=-{\frac {1}{a}}u^{n}\cos au+{\frac {n}{a}}\int u^{n-1}\cos au\,du}
I
n
=
∫
u
n
cos
a
u
d
u
=
1
a
u
n
sen
a
u
−
n
a
∫
u
n
−
1
sen
a
u
d
u
{\displaystyle I_{n}=\int u^{n}\cos au\,du={\frac {1}{a}}u^{n}\operatorname {sen} au-{\frac {n}{a}}\int u^{n-1}\operatorname {sen} au\,du}
I
n
=
∫
1
u
n
sen
a
u
d
u
=
−
1
(
n
−
1
)
u
n
−
1
sen
a
u
+
a
n
−
1
∫
1
u
n
−
1
cos
a
u
d
u
{\displaystyle I_{n}=\int {\frac {1}{u^{n}}}\operatorname {sen} au\,du=-{\frac {1}{(n-1)u^{n-1}}}\operatorname {sen} au+{\frac {a}{n-1}}\int {\frac {1}{u^{n-1}}}\cos au\,du}
I
n
=
∫
1
u
n
cos
a
u
d
u
=
−
1
(
n
−
1
)
u
n
−
1
cos
a
u
−
a
n
−
1
∫
1
u
n
−
1
sen
a
u
d
u
{\displaystyle I_{n}=\int {\frac {1}{u^{n}}}\cos au\,du=-{\frac {1}{(n-1)u^{n-1}}}\cos au-{\frac {a}{n-1}}\int {\frac {1}{u^{n-1}}}\operatorname {sen} au\,du}
I
n
=
∫
u
sen
n
a
u
d
u
=
1
a
2
n
2
(
sen
a
u
−
n
a
u
cos
a
u
)
sen
n
−
1
a
u
+
n
−
1
n
I
n
−
2
{\displaystyle I_{n}=\int u\operatorname {sen} ^{n}au\,du={\frac {1}{a^{2}n^{2}}}(\operatorname {sen} au-nau\cos au)\operatorname {sen} ^{n-1}au+{\frac {n-1}{n}}I_{n-2}}
I
n
=
∫
u
cos
n
a
u
d
u
=
1
a
2
n
2
(
cos
a
u
−
n
a
u
sen
a
u
)
cos
n
−
1
a
u
+
n
−
1
n
I
n
−
2
{\displaystyle I_{n}=\int u\cos ^{n}au\,du={\frac {1}{a^{2}n^{2}}}(\cos au-nau\operatorname {sen} au)\cos ^{n-1}au+{\frac {n-1}{n}}I_{n-2}}
I
n
=
∫
u
sec
n
a
u
d
u
=
u
(
n
−
1
)
a
sec
n
−
2
a
u
tan
a
u
−
1
(
n
−
1
)
(
n
−
2
)
a
2
sec
n
−
2
a
u
+
n
−
2
n
−
1
I
n
−
2
{\displaystyle I_{n}=\int u\sec ^{n}au\,du={\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
=
∫
u
csc
n
a
u
d
u
=
−
u
(
n
−
1
)
a
csc
n
−
2
a
u
cot
a
u
−
1
(
n
−
1
)
(
n
−
2
)
a
2
csc
n
−
2
a
u
+
n
−
2
n
−
1
I
n
−
2
{\displaystyle I_{n}=\int u\csc ^{n}au\,du=-{\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}}
I
n
=
∫
arcsin
n
u
d
u
=
(
u
arcsin
u
+
n
1
−
u
2
)
(
arcsin
u
)
n
−
1
−
n
(
n
−
1
)
I
n
−
2
{\displaystyle I_{n}=\int \arcsin ^{n}u\,du=\left(u\arcsin u+n{\sqrt {1-u^{2}}}\right)(\arcsin u)^{n-1}-n(n-1)I_{n-2}}
I
n
=
∫
arccos
n
u
d
u
=
(
u
arccos
u
−
n
1
−
u
2
)
(
arccos
u
)
n
−
1
−
n
(
n
−
1
)
I
n
−
2
{\displaystyle I_{n}=\int \arccos ^{n}u\,du=\left(u\arccos u-n{\sqrt {1-u^{2}}}\right)(\arccos u)^{n-1}-n(n-1)I_{n-2}}
I
n
=
∫
d
u
arcsin
n
u
=
u
arcsin
u
−
(
n
−
2
)
1
−
x
2
(
n
−
1
)
(
n
−
2
)
(
arcsin
u
)
n
−
1
−
1
(
n
−
1
)
(
n
−
2
)
I
n
−
2
{\displaystyle 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
=
∫
d
u
arccos
n
u
=
u
arccos
u
+
(
n
−
2
)
1
−
x
2
(
n
−
1
)
(
n
−
2
)
(
arccos
u
)
n
−
1
−
1
(
n
−
1
)
(
n
−
2
)
I
n
−
2
{\displaystyle 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
=
∫
u
n
arcsin
u
d
u
=
1
n
+
1
u
n
+
1
arcsin
u
−
1
n
+
1
∫
u
n
+
1
d
u
1
−
u
2
{\displaystyle I_{n}=\int u^{n}\arcsin u\,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
=
∫
u
n
arccos
u
d
u
=
1
n
+
1
u
n
+
1
arccos
u
+
1
n
+
1
∫
u
n
+
1
d
u
1
−
u
2
{\displaystyle I_{n}=\int u^{n}\arccos u\,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
=
∫
u
n
arcsin
u
d
u
=
1
n
+
1
u
n
(
u
arcsin
u
+
1
−
u
2
)
−
n
n
+
1
∫
u
n
−
1
1
−
u
2
d
u
{\displaystyle I_{n}=\int u^{n}\arcsin u\,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}}}\,du}
I
n
=
∫
u
n
arccos
u
d
u
=
1
n
+
1
u
n
(
u
arccos
u
−
1
−
u
2
)
+
n
n
+
1
∫
u
n
−
1
1
−
u
2
d
u
{\displaystyle I_{n}=\int u^{n}\arccos u\,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}}}\,du}
I
n
=
∫
1
u
n
arcsin
u
d
u
=
−
1
(
n
−
1
)
u
n
−
1
arcsin
u
+
1
n
−
1
∫
d
u
u
n
−
1
1
−
u
2
{\displaystyle I_{n}=\int {\frac {1}{u^{n}}}\arcsin u\,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
=
∫
1
u
n
arccos
u
d
u
=
−
1
(
n
−
1
)
u
n
−
1
arccos
u
−
1
n
−
1
∫
d
u
u
n
−
1
1
−
u
2
{\displaystyle I_{n}=\int {\frac {1}{u^{n}}}\arccos u\,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
=
∫
u
n
arctan
u
d
u
=
1
n
+
1
u
n
+
1
arctan
u
−
1
n
+
1
∫
u
n
+
1
d
u
1
+
u
2
{\displaystyle I_{n}=\int u^{n}\arctan u\,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
=
∫
u
n
arccot
u
d
u
=
1
n
+
1
u
n
+
1
arccot
u
+
1
n
+
1
∫
u
n
+
1
d
u
1
+
u
2
{\displaystyle I_{n}=\int u^{n}\operatorname {arccot} u\,du={\frac {1}{n+1}}u^{n+1}\operatorname {arccot} u+{\frac {1}{n+1}}\int {\frac {u^{n+1}du}{1+u^{2}}}}
I
n
=
∫
1
u
n
arctan
u
d
u
=
−
1
(
n
−
1
)
u
n
−
1
arctan
u
+
1
n
−
1
∫
d
u
u
n
−
1
(
1
+
u
2
)
{\displaystyle I_{n}=\int {\frac {1}{u^{n}}}\arctan u\,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
=
∫
1
u
n
arccot
u
d
u
=
−
1
(
n
−
1
)
u
n
−
1
arccot
u
−
1
n
−
1
∫
d
u
u
n
−
1
(
1
+
u
2
)
{\displaystyle I_{n}=\int {\frac {1}{u^{n}}}\operatorname {arccot} u\,du=-{\frac {1}{(n-1)u^{n-1}}}\operatorname {arccot} u-{\frac {1}{n-1}}\int {\frac {du}{u^{n-1}(1+u^{2})}}}
I
n
=
∫
u
n
e
a
u
d
u
=
1
a
u
n
e
a
u
−
n
a
I
n
−
1
{\displaystyle I_{n}=\int u^{n}e^{au}du={\frac {1}{a}}u^{n}e^{au}-{\frac {n}{a}}I_{n-1}}
I
n
=
∫
1
u
n
e
a
u
d
u
=
−
1
(
n
−
1
)
u
n
−
1
e
a
u
−
a
n
−
1
I
n
−
1
{\displaystyle 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
=
∫
u
n
e
a
u
2
d
u
=
1
2
a
u
n
−
1
e
a
u
2
−
n
−
1
2
a
I
n
−
2
{\displaystyle I_{n}=\int u^{n}e^{au^{2}}du={\frac {1}{2a}}u^{n-1}e^{au^{2}}-{\frac {n-1}{2a}}I_{n-2}}
I
n
=
∫
e
a
u
sen
n
b
u
d
u
=
1
a
2
+
b
2
n
2
e
a
u
(
a
sen
b
u
−
n
b
cos
b
u
)
sen
n
−
1
b
u
+
n
(
n
−
1
)
b
2
a
2
+
b
2
n
2
I
n
−
2
{\displaystyle I_{n}=\int e^{au}\operatorname {sen} ^{n}bu\,du={\frac {1}{a^{2}+b^{2}n^{2}}}e^{au}(a\operatorname {sen} bu-nb\cos bu)\operatorname {sen} ^{n-1}bu+{\frac {n(n-1)b^{2}}{a^{2}+b^{2}n^{2}}}I_{n-2}}
I
n
=
∫
e
a
u
cos
n
b
u
d
u
=
1
a
2
+
b
2
n
2
e
a
u
(
a
cos
b
u
−
n
b
sen
b
u
)
cos
n
−
1
b
u
+
n
(
n
−
1
)
b
2
a
2
+
b
2
n
2
I
n
−
2
{\displaystyle I_{n}=\int e^{au}\cos ^{n}bu\,du={\frac {1}{a^{2}+b^{2}n^{2}}}e^{au}(a\cos bu-nb\operatorname {sen} bu)\cos ^{n-1}bu+{\frac {n(n-1)b^{2}}{a^{2}+b^{2}n^{2}}}I_{n-2}}
I
n
=
∫
x
n
e
−
x
d
x
=
−
e
−
x
[
∑
k
=
0
n
d
k
(
x
n
)
d
x
k
]
;
d
0
(
x
n
)
d
x
0
=
x
n
;
d
n
(
x
n
)
d
x
n
=
n
!
{\displaystyle {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!}
I
n
=
∫
ln
n
u
d
u
=
u
ln
n
u
−
n
I
n
−
1
{\displaystyle I_{n}=\int \ln ^{n}u\,du=u\ln ^{n}u-nI_{n-1}}
I
n
=
∫
d
u
ln
n
u
=
−
u
(
n
−
1
)
ln
n
−
1
u
+
1
n
−
1
I
n
−
1
{\displaystyle 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
=
∫
u
m
ln
n
u
d
u
=
1
m
+
1
u
m
+
1
ln
n
u
−
n
m
+
1
I
n
−
1
{\displaystyle I_{n}=\int u^{m}\ln ^{n}u\,du={\frac {1}{m+1}}u^{m+1}\ln ^{n}u-{\frac {n}{m+1}}I_{n-1}}
I
n
=
∫
1
u
m
ln
n
u
d
u
=
−
1
(
m
−
1
)
u
m
−
1
ln
n
u
+
n
m
−
1
I
n
−
1
{\displaystyle I_{n}=\int {\frac {1}{u^{m}}}\ln ^{n}u\,du=-{\frac {1}{(m-1)u^{m-1}}}\ln ^{n}u+{\frac {n}{m-1}}I_{n-1}}
I
n
=
∫
u
m
ln
n
u
d
u
=
−
u
m
+
1
(
n
−
1
)
ln
n
−
1
u
+
m
+
1
n
−
1
I
n
−
1
{\displaystyle 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
=
∫
d
u
u
m
ln
n
u
=
−
1
(
n
−
1
)
u
m
−
1
ln
n
−
1
u
−
m
−
1
n
−
1
I
n
−
1
{\displaystyle 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}}
I
n
=
∫
sinh
n
u
d
u
=
1
n
sinh
n
−
1
u
cosh
u
−
n
−
1
n
I
n
−
2
{\displaystyle I_{n}=\int \sinh ^{n}u\,du={\frac {1}{n}}\sinh ^{n-1}u\,\cosh u-{\frac {n-1}{n}}I_{n-2}}
I
n
=
∫
cosh
n
u
d
u
=
1
n
cosh
n
−
1
u
sinh
u
+
n
−
1
n
I
n
−
2
{\displaystyle I_{n}=\int \cosh ^{n}u\,du={\frac {1}{n}}\cosh ^{n-1}u\,\sinh u+{\frac {n-1}{n}}I_{n-2}}
I
n
=
∫
s
e
c
h
n
u
d
u
=
1
n
−
1
s
e
c
h
n
−
2
u
tanh
u
+
n
−
2
n
−
1
I
n
−
2
{\displaystyle I_{n}=\int \mathrm {sech^{n}u} \,du={\frac {1}{n-1}}\mathrm {sech^{n-2}u} \,\tanh u+{\frac {n-2}{n-1}}I_{n-2}}
I
n
=
∫
c
s
c
h
n
u
d
u
=
−
1
n
−
1
c
s
c
h
n
−
2
u
coth
u
−
n
−
2
n
−
1
I
n
−
2
{\displaystyle I_{n}=\int \mathrm {csch^{n}u} \,du=-{\frac {1}{n-1}}\mathrm {csch^{n-2}u} \,\coth u-{\frac {n-2}{n-1}}I_{n-2}}
I
n
=
∫
tanh
n
u
d
u
=
−
1
n
−
1
tanh
n
−
1
u
+
I
n
−
2
{\displaystyle I_{n}=\int \tanh ^{n}u\,du=-{\frac {1}{n-1}}\tanh ^{n-1}u+I_{n-2}}
I
n
=
∫
coth
n
u
d
u
=
−
1
n
−
1
coth
n
−
1
u
+
I
n
−
2
{\displaystyle I_{n}=\int \coth ^{n}u\,du=-{\frac {1}{n-1}}\coth ^{n-1}u+I_{n-2}}
I
m
,
n
=
∫
sinh
m
u
cosh
n
u
d
u
=
1
m
+
n
sinh
m
−
1
u
cosh
n
+
1
u
−
m
−
1
m
+
n
I
m
−
2
,
n
{\displaystyle I_{m,n}=\int \sinh ^{m}u\,\cosh ^{n}u\,du={\frac {1}{m+n}}\sinh ^{m-1}u\,\cosh ^{n+1}u-{\frac {m-1}{m+n}}I_{m-2,n}}
I
m
,
n
=
∫
sinh
m
u
cosh
n
u
d
u
=
1
m
+
n
sinh
m
+
1
u
cosh
n
−
1
u
+
n
−
1
m
+
n
I
m
,
n
−
2
{\displaystyle I_{m,n}=\int \sinh ^{m}u\,\cosh ^{n}u\,du={\frac {1}{m+n}}\sinh ^{m+1}u\,\cosh ^{n-1}u+{\frac {n-1}{m+n}}I_{m,n-2}}
I
m
,
n
=
∫
sinh
m
u
cosh
n
u
d
u
=
−
sinh
m
−
1
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{\displaystyle 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}}
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{\displaystyle 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}}
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{\displaystyle I_{n}=\int u^{n}\sinh au\,du={\frac {1}{a}}u^{n}\cosh au-{\frac {n}{a}}\int u^{n-1}\cosh au\,du}
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{\displaystyle I_{n}=\int u^{n}\cosh au\,du={\frac {1}{a}}u^{n}\sinh au-{\frac {n}{a}}\int u^{n-1}\sinh au\,du}
![{\displaystyle {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!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9cc39cac876b15e8fc8338fe3ed48d6d8db539)
