editar
La fórmula de reducción puede ser obtenida utilizando los métodos de integración más comunes tales como integración por sustitución, integración por partes , integración por sustitución trigonométrica, integración por fracciones parciales , etc. La idea principal consiste en expresar un integral que contiene un parámetro entero de una función, representado por
I
n
{\displaystyle I_{n}}
I
n
−
1
{\displaystyle I_{n-1}}
I
n
−
2
{\displaystyle I_{n-2}}
relación de recurrencia . En otras palabras, la fórmula de reducción expresa la integral
I
n
=
∫
f
(
x
,
n
)
d
x
,
{\displaystyle I_{n}=\int f(x,n)\,{\text{d}}x,}
en términos de
I
k
=
∫
f
(
x
,
k
)
d
x
,
{\displaystyle I_{k}=\int f(x,k)\,{\text{d}}x,}
donde
k
<
n
.
{\displaystyle k<n.}
Para evaluar la integral, comenzamos por nombrar la integral como
I
n
{\displaystyle I_{n}}
I
n
−
1
{\displaystyle I_{n-1}}
I
n
−
2
{\displaystyle I_{n-2}}
I
{\displaystyle I}
I
{\displaystyle I}
I
n
{\displaystyle I_{n}}
[ 1]
Abajo se muestran ejemplos del procedimiento.
Típicamente, integrales como
∫
cos
n
(
x
)
d
x
{\displaystyle \int \cos ^{n}(x)\,dx}
pueden ser evaluadas por una fórmula de reducción.
Empezamos por nombrar:
I
n
=
∫
cos
n
(
x
)
d
x
.
{\displaystyle I_{n}=\int \cos ^{n}(x)\,dx.}
la reescribimos como
I
n
=
∫
cos
n
−
1
(
x
)
cos
(
x
)
d
x
{\displaystyle I_{n}=\int \cos ^{n-1}(x)\cos(x)\,dx}
integrando por la sustitución:
cos
x
d
x
=
d
(
sen
x
)
{\displaystyle \cos x\,dx=d(\operatorname {sen} x)}
I
n
=
∫
cos
n
−
1
(
x
)
d
(
sen
x
)
{\displaystyle I_{n}=\int \cos ^{n-1}(x)\,d(\operatorname {sen} x)}
integrando por partes:
∫
cos
n
(
x
)
d
x
=
cos
n
−
1
(
x
)
sen
x
−
∫
sen
x
d
(
cos
n
−
1
(
x
)
)
=
cos
n
−
1
(
x
)
sen
x
+
(
n
−
1
)
∫
sen
x
cos
n
−
2
(
x
)
sen
x
d
x
=
cos
n
−
1
(
x
)
sen
x
+
(
n
−
1
)
∫
cos
n
−
2
(
x
)
sen
2
(
x
)
d
x
=
cos
n
−
1
(
x
)
sen
x
+
(
n
−
1
)
∫
cos
n
−
2
(
x
)
(
1
−
cos
2
(
x
)
)
d
x
=
cos
n
−
1
(
x
)
sen
x
+
(
n
−
1
)
∫
cos
n
−
2
(
x
)
d
x
−
(
n
−
1
)
∫
cos
n
(
x
)
d
x
=
cos
n
−
1
(
x
)
sen
x
+
(
n
−
1
)
I
n
−
2
−
(
n
−
1
)
I
n
,
{\displaystyle {\begin{aligned}\int \cos ^{n}(x)\,dx&=\cos ^{n-1}(x)\operatorname {sen} x-\int \operatorname {sen} x\,d(\cos ^{n-1}(x))\\&=\cos ^{n-1}(x)\operatorname {sen} x+(n-1)\int \operatorname {sen} x\cos ^{n-2}(x)\operatorname {sen} x\,dx\\&=\cos ^{n-1}(x)\operatorname {sen} x+(n-1)\int \cos ^{n-2}(x)\operatorname {sen} ^{2}(x)\,dx\\&=\cos ^{n-1}(x)\operatorname {sen} x+(n-1)\int \cos ^{n-2}(x)(1-\cos ^{2}(x))\,dx\\&=\cos ^{n-1}(x)\operatorname {sen} x+(n-1)\int \cos ^{n-2}(x)\,dx-(n-1)\int \cos ^{n}(x)\,dx\\&=\cos ^{n-1}(x)\operatorname {sen} x+(n-1)I_{n-2}-(n-1)I_{n},\end{aligned}}\,}
resolviendo para
I
n
{\displaystyle I_{n}}
I
n
+
(
n
−
1
)
I
n
=
cos
n
−
1
(
x
)
sen
x
+
(
n
−
1
)
I
n
−
2
n
I
n
=
cos
n
−
1
(
x
)
sen
x
+
(
n
−
1
)
I
n
−
2
I
n
=
1
n
cos
n
−
1
(
x
)
sen
x
+
n
−
1
n
I
n
−
2
{\displaystyle {\begin{aligned}&I_{n}+(n-1)I_{n}=\cos ^{n-1}(x)\operatorname {sen} x+(n-1)I_{n-2}\\&nI_{n}\ =\cos ^{n-1}(x)\operatorname {sen} x\ +(n-1)I_{n-2}\\&I_{n}\ ={\frac {1}{n}}\cos ^{n-1}(x)\operatorname {sen} x\ +{\frac {n-1}{n}}I_{n-2}\end{aligned}}}
por lo que la fórmula de reducción es:
∫
cos
n
(
x
)
d
x
=
cos
n
−
1
(
x
)
sen
(
x
)
n
+
n
−
1
n
∫
cos
n
−
2
(
x
)
d
x
{\displaystyle \int \cos ^{n}(x)\,dx={\frac {\cos ^{n-1}(x)\operatorname {sen}(x)}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}(x)\,dx}
Por ejemplo, podemos utilizar la fórmula anterior para evaluar la integral para
n
=
5
{\displaystyle n=5}
I
5
=
∫
cos
5
(
x
)
d
x
{\displaystyle I_{5}=\int \cos ^{5}(x)\,dx}
Calculando los índices
n
=
5
,
I
5
=
1
5
cos
4
x
sen
x
+
4
5
I
3
,
{\displaystyle n=5,\quad I_{5}={\tfrac {1}{5}}\cos ^{4}x\operatorname {sen} x+{\tfrac {4}{5}}I_{3},\,}
n
=
3
,
I
3
=
1
3
cos
2
x
sen
x
+
2
3
I
1
,
{\displaystyle n=3,\quad I_{3}={\tfrac {1}{3}}\cos ^{2}x\operatorname {sen} x+{\tfrac {2}{3}}I_{1},\,}
sustituyendo “hacia atrás”:
∵
I
1
=
∫
cos
x
d
x
=
sen
x
+
C
1
,
{\displaystyle \because I_{1}\ =\int \cos x\,{\text{d}}x=\operatorname {sen} x+C_{1},\,}
∴
I
3
=
1
3
cos
2
x
sen
x
+
2
3
sen
x
+
C
2
,
C
2
=
2
3
C
1
,
{\displaystyle \therefore I_{3}\ ={\tfrac {1}{3}}\cos ^{2}x\operatorname {sen} x+{\tfrac {2}{3}}\operatorname {sen} x+C_{2},\quad C_{2}\ ={\tfrac {2}{3}}C_{1},\,}
por lo tanto
I
5
=
∫
cos
5
(
x
)
d
x
=
1
5
cos
4
(
x
)
sen
x
+
4
5
[
1
3
cos
2
(
x
)
sen
x
+
2
3
sen
x
]
+
C
,
{\displaystyle I_{5}=\int \cos ^{5}(x)\;dx={\frac {1}{5}}\cos ^{4}(x)\operatorname {sen} x+{\frac {4}{5}}\left[{\frac {1}{3}}\cos ^{2}(x)\operatorname {sen} x+{\frac {2}{3}}\operatorname {sen} x\right]+C,\,}
donde
C
∈
R
{\displaystyle C\in \mathbb {R} }
Otro ejemplo típico es:
∫
x
n
e
a
x
d
x
{\displaystyle \int x^{n}e^{ax}\;dx}
Iniciamos por nombrar:
I
n
=
∫
x
n
e
a
x
d
x
{\displaystyle I_{n}=\int x^{n}e^{ax}\,dx}
integrando por sustitución:
x
n
d
x
=
d
(
x
n
+
1
)
n
+
1
,
{\displaystyle x^{n}\,dx={\frac {d(x^{n+1})}{n+1}},\,\!}
I
n
=
1
n
+
1
∫
e
a
x
d
(
x
n
+
1
)
{\displaystyle I_{n}={\frac {1}{n+1}}\int e^{ax}\,d(x^{n+1})}
Ahora integrando por partes:
∫
e
a
x
d
(
x
n
+
1
)
=
x
n
+
1
e
a
x
−
∫
x
n
+
1
d
(
e
a
x
)
=
x
n
+
1
e
a
x
−
a
∫
x
n
+
1
e
a
x
d
x
{\displaystyle {\begin{aligned}\int e^{ax}\,d(x^{n+1})&=x^{n+1}e^{ax}-\int x^{n+1}\,d(e^{ax})\\&=x^{n+1}e^{ax}-a\int x^{n+1}e^{ax}\,dx\end{aligned}}}
(
n
+
1
)
I
n
=
x
n
+
1
e
a
x
−
a
I
n
+
1
,
{\displaystyle (n+1)I_{n}=x^{n+1}e^{ax}-aI_{n+1},\!}
recorriendo los índices (esto es
n
+
1
→
n
{\displaystyle n+1\to n}
n
→
n
−
1
{\displaystyle n\to n-1}
n
I
n
−
1
=
x
n
e
a
x
−
a
I
n
,
{\displaystyle nI_{n-1}=x^{n}e^{ax}-aI_{n},\!}
resolviendo para
I
n
{\displaystyle I_{n}}
I
n
=
1
a
(
x
n
e
a
x
−
n
I
n
−
1
)
,
{\displaystyle I_{n}={\frac {1}{a}}\left(x^{n}e^{ax}-nI_{n-1}\right),\,\!}
por lo que la fórmula de reducción es:
∫
x
n
e
a
x
d
x
=
1
a
(
x
n
e
a
x
−
n
∫
x
n
−
1
e
a
x
d
x
)
.
{\displaystyle \int x^{n}e^{ax}\,dx={\frac {1}{a}}\left(x^{n}e^{ax}-n\int x^{n-1}e^{ax}\,dx\right).}
Otra manera en que se pudo obtener la fórmula anterior pudo haber sido sustituyendo en un principio
e
a
x
{\displaystyle e^{ax}}
Integración por sustitución:
e
a
x
d
x
=
d
(
e
a
x
)
a
{\displaystyle e^{ax}\,dx={\frac {d(e^{ax})}{a}}}
I
n
=
1
a
∫
x
n
d
(
e
a
x
)
{\displaystyle I_{n}={\frac {1}{a}}\int x^{n}\,d(e^{ax})}
Ahora integrando por partes:
∫
x
n
d
(
e
a
x
)
=
x
n
e
a
x
−
∫
e
a
x
d
(
x
n
)
=
x
n
e
a
x
−
n
∫
e
a
x
x
n
−
1
d
x
,
{\displaystyle {\begin{aligned}\int x^{n}\,d(e^{ax})&=x^{n}e^{ax}-\int e^{ax}\,d(x^{n})\\&=x^{n}e^{ax}-n\int e^{ax}x^{n-1}\,dx,\end{aligned}}}
que da la fórmula de reducción cuando “sustituye hacia atrás”:
I
n
=
1
a
(
x
n
e
a
x
−
n
I
n
−
1
)
,
{\displaystyle I_{n}={\frac {1}{a}}\left(x^{n}e^{ax}-nI_{n-1}\right),\,\!}
que es equivalente a:
∫
x
n
e
a
x
d
x
=
1
a
(
x
n
e
a
x
−
n
∫
x
n
−
1
e
a
x
d
x
)
{\displaystyle \int x^{n}e^{ax}\,dx={\frac {1}{a}}\left(x^{n}e^{ax}-n\int x^{n-1}e^{ax}\,dx\right)}
editar
Las siguientes integrales contienen:[ 2]
Factores del radical lineal
a
x
+
b
{\displaystyle {\sqrt {ax+b}}}
Factores lineales
p
x
+
q
{\displaystyle {px+q}\,\!}
a
x
+
b
{\displaystyle {\sqrt {ax+b}}}
Factores cuadráticos
x
2
+
a
2
{\displaystyle x^{2}+a^{2}}
Factores cuadráticos
x
2
−
a
2
{\displaystyle x^{2}-a^{2}\,\!}
x
>
a
{\displaystyle x>a}
Factores cuadráticos
a
2
−
x
2
{\displaystyle a^{2}-x^{2}\,\!}
x
<
a
{\displaystyle x<a}
(Irreductible ) factores cuadráticos
a
x
2
+
b
x
+
c
{\displaystyle ax^{2}+bx+c}
Radicales de factores cuadráticos irreductibles
a
x
2
+
b
x
+
c
{\displaystyle {\sqrt {ax^{2}+bx+c}}}
Integral
Fórmula de reducción
I
n
=
∫
x
n
a
x
+
b
d
x
{\displaystyle I_{n}=\int {\frac {x^{n}}{\sqrt {ax+b}}}\,{\text{d}}x\,\!}
I
n
=
2
x
n
a
x
+
b
a
(
2
n
+
1
)
−
2
n
b
a
(
2
n
+
1
)
I
n
−
1
{\displaystyle I_{n}={\frac {2x^{n}{\sqrt {ax+b}}}{a(2n+1)}}-{\frac {2nb}{a(2n+1)}}I_{n-1}\,\!}
I
n
=
∫
d
x
x
n
a
x
+
b
{\displaystyle I_{n}=\int {\frac {{\text{d}}x}{x^{n}{\sqrt {ax+b}}}}\,\!}
I
n
=
−
a
x
+
b
(
n
−
1
)
b
x
n
−
1
−
a
(
2
n
−
3
)
2
b
(
n
−
1
)
I
n
−
1
{\displaystyle I_{n}=-{\frac {\sqrt {ax+b}}{(n-1)bx^{n-1}}}-{\frac {a(2n-3)}{2b(n-1)}}I_{n-1}\,\!}
I
n
=
∫
x
n
a
x
+
b
d
x
{\displaystyle I_{n}=\int x^{n}{\sqrt {ax+b}}\,{\text{d}}x\,\!}
I
n
=
2
x
n
(
a
x
+
b
)
3
a
(
2
n
+
3
)
−
2
n
b
a
(
2
n
+
3
)
I
n
−
1
{\displaystyle I_{n}={\frac {2x^{n}{\sqrt {(ax+b)^{3}}}}{a(2n+3)}}-{\frac {2nb}{a(2n+3)}}I_{n-1}\,\!}
I
m
,
n
=
∫
d
x
(
a
x
+
b
)
m
(
p
x
+
q
)
n
{\displaystyle I_{m,n}=\int {\frac {{\text{d}}x}{(ax+b)^{m}(px+q)^{n}}}\,\!}
I
m
,
n
=
{
−
1
(
n
−
1
)
(
b
p
−
a
q
)
[
1
(
a
x
+
b
)
m
−
1
(
p
x
+
q
)
n
−
1
+
a
(
m
+
n
−
2
)
I
m
,
n
−
1
]
1
(
m
−
1
)
(
b
p
−
a
q
)
[
1
(
a
x
+
b
)
m
−
1
(
p
x
+
q
)
n
−
1
+
p
(
m
+
n
−
2
)
I
m
−
1
,
n
]
{\displaystyle I_{m,n}={\begin{cases}-{\frac {1}{(n-1)(bp-aq)}}\left[{\frac {1}{(ax+b)^{m-1}(px+q)^{n-1}}}+a(m+n-2)I_{m,n-1}\right]\\{\frac {1}{(m-1)(bp-aq)}}\left[{\frac {1}{(ax+b)^{m-1}(px+q)^{n-1}}}+p(m+n-2)I_{m-1,n}\right]\end{cases}}\,\!}
I
m
,
n
=
∫
(
a
x
+
b
)
m
(
p
x
+
q
)
n
d
x
{\displaystyle I_{m,n}=\int {\frac {(ax+b)^{m}}{(px+q)^{n}}}\,{\text{d}}x\,\!}
I
m
,
n
=
{
−
1
(
n
−
1
)
(
b
p
−
a
q
)
[
(
a
x
+
b
)
m
+
1
(
p
x
+
q
)
n
−
1
+
a
(
n
−
m
−
2
)
I
m
,
n
−
1
]
−
1
(
n
−
m
−
1
)
p
[
(
a
x
+
b
)
m
(
p
x
+
q
)
n
−
1
+
m
(
b
p
−
a
q
)
I
m
−
1
,
n
]
−
1
(
n
−
1
)
p
[
(
a
x
+
b
)
m
(
p
x
+
q
)
n
−
1
−
a
m
I
m
−
1
,
n
−
1
]
{\displaystyle I_{m,n}={\begin{cases}-{\frac {1}{(n-1)(bp-aq)}}\left[{\frac {(ax+b)^{m+1}}{(px+q)^{n-1}}}+a(n-m-2)I_{m,n-1}\right]\\-{\frac {1}{(n-m-1)p}}\left[{\frac {(ax+b)^{m}}{(px+q)^{n-1}}}+m(bp-aq)I_{m-1,n}\right]\\-{\frac {1}{(n-1)p}}\left[{\frac {(ax+b)^{m}}{(px+q)^{n-1}}}-amI_{m-1,n-1}\right]\end{cases}}\,\!}
Integral
Fórmula de reducción
I
n
=
∫
d
x
(
x
2
+
a
2
)
n
{\displaystyle I_{n}=\int {\frac {{\text{d}}x}{(x^{2}+a^{2})^{n}}}\,\!}
I
n
=
x
2
a
2
(
n
−
1
)
(
x
2
+
a
2
)
n
−
1
+
2
n
−
3
2
a
2
(
n
−
1
)
I
n
−
1
{\displaystyle I_{n}={\frac {x}{2a^{2}(n-1)(x^{2}+a^{2})^{n-1}}}+{\frac {2n-3}{2a^{2}(n-1)}}I_{n-1}\,\!}
I
n
,
m
=
∫
d
x
x
m
(
x
2
+
a
2
)
n
{\displaystyle I_{n,m}=\int {\frac {{\text{d}}x}{x^{m}(x^{2}+a^{2})^{n}}}\,\!}
a
2
I
n
,
m
=
I
m
,
n
−
1
−
I
m
−
2
,
n
{\displaystyle a^{2}I_{n,m}=I_{m,n-1}-I_{m-2,n}\,\!}
I
n
,
m
=
∫
x
m
(
x
2
+
a
2
)
n
d
x
{\displaystyle I_{n,m}=\int {\frac {x^{m}}{(x^{2}+a^{2})^{n}}}\,{\text{d}}x\,\!}
I
n
,
m
=
I
m
−
2
,
n
−
1
−
a
2
I
m
−
2
,
n
{\displaystyle I_{n,m}=I_{m-2,n-1}-a^{2}I_{m-2,n}\,\!}
Integral
Fórmula de reducción
I
n
=
∫
d
x
(
x
2
−
a
2
)
n
{\displaystyle I_{n}=\int {\frac {{\text{d}}x}{(x^{2}-a^{2})^{n}}}\,\!}
I
n
=
−
x
2
a
2
(
n
−
1
)
(
x
2
−
a
2
)
n
−
1
−
2
n
−
3
2
a
2
(
n
−
1
)
I
n
−
1
{\displaystyle I_{n}=-{\frac {x}{2a^{2}(n-1)(x^{2}-a^{2})^{n-1}}}-{\frac {2n-3}{2a^{2}(n-1)}}I_{n-1}\,\!}
I
n
,
m
=
∫
d
x
x
m
(
x
2
−
a
2
)
n
{\displaystyle I_{n,m}=\int {\frac {{\text{d}}x}{x^{m}(x^{2}-a^{2})^{n}}}\,\!}
a
2
I
n
,
m
=
I
m
−
2
,
n
−
I
m
,
n
−
1
{\displaystyle {a^{2}}I_{n,m}=I_{m-2,n}-I_{m,n-1}\,\!}
I
n
,
m
=
∫
x
m
(
x
2
−
a
2
)
n
d
x
{\displaystyle I_{n,m}=\int {\frac {x^{m}}{(x^{2}-a^{2})^{n}}}\,{\text{d}}x\,\!}
I
n
,
m
=
I
m
−
2
,
n
−
1
+
a
2
I
m
−
2
,
n
{\displaystyle I_{n,m}=I_{m-2,n-1}+a^{2}I_{m-2,n}\,\!}
Integral
Fórmula de reducción
I
n
=
∫
d
x
(
a
2
−
x
2
)
n
{\displaystyle I_{n}=\int {\frac {{\text{d}}x}{(a^{2}-x^{2})^{n}}}\,\!}
I
n
=
x
2
a
2
(
n
−
1
)
(
a
2
−
x
2
)
n
−
1
+
2
n
−
3
2
a
2
(
n
−
1
)
I
n
−
1
{\displaystyle I_{n}={\frac {x}{2a^{2}(n-1)(a^{2}-x^{2})^{n-1}}}+{\frac {2n-3}{2a^{2}(n-1)}}I_{n-1}\,\!}
I
n
,
m
=
∫
d
x
x
m
(
a
2
−
x
2
)
n
{\displaystyle I_{n,m}=\int {\frac {{\text{d}}x}{x^{m}(a^{2}-x^{2})^{n}}}\,\!}
a
2
I
n
,
m
=
I
m
,
n
−
1
+
I
m
−
2
,
n
{\displaystyle {a^{2}}I_{n,m}=I_{m,n-1}+I_{m-2,n}\,\!}
I
n
,
m
=
∫
x
m
(
a
2
−
x
2
)
n
d
x
{\displaystyle I_{n,m}=\int {\frac {x^{m}}{(a^{2}-x^{2})^{n}}}\,{\text{d}}x\,\!}
I
n
,
m
=
a
2
I
m
−
2
,
n
−
I
m
−
2
,
n
−
1
{\displaystyle I_{n,m}=a^{2}I_{m-2,n}-I_{m-2,n-1}\,\!}
Integral
Fórmula de reducción
I
n
=
∫
d
x
x
n
(
a
x
2
+
b
x
+
c
)
{\displaystyle I_{n}=\int {\frac {{\text{d}}x}{{x^{n}}(ax^{2}+bx+c)}}\,\!}
−
c
I
n
=
1
x
n
−
1
(
n
−
1
)
+
b
I
n
−
1
+
a
I
n
−
2
{\displaystyle -cI_{n}={\frac {1}{x^{n-1}(n-1)}}+bI_{n-1}+aI_{n-2}\,\!}
I
m
,
n
=
∫
x
m
d
x
(
a
x
2
+
b
x
+
c
)
n
{\displaystyle I_{m,n}=\int {\frac {x^{m}\,{\text{d}}x}{(ax^{2}+bx+c)^{n}}}\,\!}
I
m
,
n
=
−
x
m
−
1
a
(
2
n
−
m
−
1
)
(
a
x
2
+
b
x
+
c
)
n
−
1
−
b
(
n
−
m
)
a
(
2
n
−
m
−
1
)
I
m
−
1
,
n
+
c
(
m
−
1
)
a
(
2
n
−
m
−
1
)
I
m
−
2
,
n
{\displaystyle I_{m,n}=-{\frac {x^{m-1}}{a(2n-m-1)(ax^{2}+bx+c)^{n-1}}}-{\frac {b(n-m)}{a(2n-m-1)}}I_{m-1,n}+{\frac {c(m-1)}{a(2n-m-1)}}I_{m-2,n}\,\!}
I
m
,
n
=
∫
d
x
x
m
(
a
x
2
+
b
x
+
c
)
n
{\displaystyle I_{m,n}=\int {\frac {{\text{d}}x}{x^{m}(ax^{2}+bx+c)^{n}}}\,\!}
−
c
(
m
−
1
)
I
m
,
n
=
1
x
m
−
1
(
a
x
2
+
b
x
+
c
)
n
−
1
+
a
(
m
+
2
n
−
3
)
I
m
−
2
,
n
+
b
(
m
+
n
−
2
)
I
m
−
1
,
n
{\displaystyle -c(m-1)I_{m,n}={\frac {1}{x^{m-1}(ax^{2}+bx+c)^{n-1}}}+{a(m+2n-3)}I_{m-2,n}+{b(m+n-2)}I_{m-1,n}\,\!}
Integral
Fórmula de reducción
I
n
=
∫
(
a
x
2
+
b
x
+
c
)
n
d
x
{\displaystyle I_{n}=\int (ax^{2}+bx+c)^{n}\,{\text{d}}x\,\!}
8
a
(
n
+
1
)
I
n
+
1
2
=
2
(
2
a
x
+
b
)
(
a
x
2
+
b
x
+
c
)
n
+
1
2
+
(
2
n
+
1
)
(
4
a
c
−
b
2
)
I
n
−
1
2
{\displaystyle 8a(n+1)I_{n+{\frac {1}{2}}}=2(2ax+b)(ax^{2}+bx+c)^{n+{\frac {1}{2}}}+(2n+1)(4ac-b^{2})I_{n-{\frac {1}{2}}}\,\!}
I
n
=
∫
1
(
a
x
2
+
b
x
+
c
)
n
d
x
{\displaystyle I_{n}=\int {\frac {1}{(ax^{2}+bx+c)^{n}}}\,{\text{d}}x\,\!}
(
2
n
−
1
)
(
4
a
c
−
b
2
)
I
n
+
1
2
=
2
(
2
a
x
+
b
)
(
a
x
2
+
b
x
+
c
)
n
−
1
2
+
8
a
(
n
−
1
)
I
n
−
1
2
{\displaystyle (2n-1)(4ac-b^{2})I_{n+{\frac {1}{2}}}={\frac {2(2ax+b)}{(ax^{2}+bx+c)^{n-{\frac {1}{2}}}}}+{8a(n-1)}I_{n-{\frac {1}{2}}}\,\!}
Nótese que por los leyes de los exponentes :
I
n
+
1
2
=
I
2
n
+
1
2
=
∫
1
(
a
x
2
+
b
x
+
c
)
2
n
+
1
2
d
x
=
∫
1
(
a
x
2
+
b
x
+
c
)
2
n
+
1
d
x
{\displaystyle I_{n+{\frac {1}{2}}}=I_{\frac {2n+1}{2}}=\int {\frac {1}{(ax^{2}+bx+c)^{\frac {2n+1}{2}}}}\,{\text{d}}x=\int {\frac {1}{\sqrt {(ax^{2}+bx+c)^{2n+1}}}}\,{\text{d}}x\,\!}
Funciones trascendentes
editar
Las siguientes integrales contienen:[ 2]
Factores de seno
Factores de coseno
Factores de productos o cocientes de seno y coseno
Cocientes/productos de factores exponenciales y potencias de x
Productos de exponenciales y factores de seno/coseno
Integral
Fórmula de reducción
I
n
=
∫
x
n
sen
a
x
d
x
{\displaystyle I_{n}=\int x^{n}\operatorname {sen} {ax}\,{\text{d}}x\,\!}
a
2
I
n
=
−
a
x
n
cos
a
x
+
n
x
n
−
1
sen
a
x
−
n
(
n
−
1
)
I
n
−
2
{\displaystyle a^{2}I_{n}=-ax^{n}\cos {ax}+nx^{n-1}\operatorname {sen} {ax}-n(n-1)I_{n-2}\,\!}
J
n
=
∫
x
n
cos
a
x
d
x
{\displaystyle J_{n}=\int x^{n}\cos {ax}\,{\text{d}}x\,\!}
a
2
J
n
=
a
x
n
sen
a
x
+
n
x
n
−
1
cos
a
x
−
n
(
n
−
1
)
J
n
−
2
{\displaystyle a^{2}J_{n}=ax^{n}\operatorname {sen} {ax}+nx^{n-1}\cos {ax}-n(n-1)J_{n-2}\,\!}
I
n
=
∫
sen
a
x
x
n
d
x
{\displaystyle I_{n}=\int {\frac {\operatorname {sen} {ax}}{x^{n}}}\,{\text{d}}x\,\!}
J
n
=
∫
cos
a
x
x
n
d
x
{\displaystyle J_{n}=\int {\frac {\cos {ax}}{x^{n}}}\,{\text{d}}x\,\!}
I
n
=
−
sen
a
x
(
n
−
1
)
x
n
−
1
+
a
n
−
1
J
n
−
1
{\displaystyle I_{n}=-{\frac {\operatorname {sen} {ax}}{(n-1)x^{n-1}}}+{\frac {a}{n-1}}J_{n-1}\,\!}
J
n
=
−
cos
a
x
(
n
−
1
)
x
n
−
1
−
a
n
−
1
I
n
−
1
{\displaystyle J_{n}=-{\frac {\cos {ax}}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}I_{n-1}\,\!}
El formulae puede ser combinado para obtener ecuaciones separadas en En:
J
n
−
1
=
−
cos
a
x
(
n
−
2
)
x
n
−
2
−
a
n
−
2
I
n
−
2
{\displaystyle J_{n-1}=-{\frac {\cos {ax}}{(n-2)x^{n-2}}}-{\frac {a}{n-2}}I_{n-2}\,\!}
I
n
=
−
sen
a
x
(
n
−
1
)
x
n
−
1
−
a
n
−
1
[
cos
a
x
(
n
−
2
)
x
n
−
2
+
a
n
−
2
I
n
−
2
]
{\displaystyle I_{n}=-{\frac {\operatorname {sen} {ax}}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}\left[{\frac {\cos {ax}}{(n-2)x^{n-2}}}+{\frac {a}{n-2}}I_{n-2}\right]\,\!}
∴
I
n
=
−
sen
a
x
(
n
−
1
)
x
n
−
1
−
a
(
n
−
1
)
(
n
−
2
)
(
cos
a
x
x
n
−
2
+
a
I
n
−
2
)
{\displaystyle \therefore I_{n}=-{\frac {\operatorname {sen} {ax}}{(n-1)x^{n-1}}}-{\frac {a}{(n-1)(n-2)}}\left({\frac {\cos {ax}}{x^{n-2}}}+aI_{n-2}\right)\,\!}
Y Jn:
I
n
−
1
=
−
sen
a
x
(
n
−
2
)
x
n
−
2
+
a
n
−
2
J
n
−
2
{\displaystyle I_{n-1}=-{\frac {\operatorname {sen} {ax}}{(n-2)x^{n-2}}}+{\frac {a}{n-2}}J_{n-2}\,\!}
J
n
=
−
cos
a
x
(
n
−
1
)
x
n
−
1
−
a
n
−
1
[
−
sen
a
x
(
n
−
2
)
x
n
−
2
+
a
n
−
2
J
n
−
2
]
{\displaystyle J_{n}=-{\frac {\cos {ax}}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}\left[-{\frac {\operatorname {sen} {ax}}{(n-2)x^{n-2}}}+{\frac {a}{n-2}}J_{n-2}\right]\,\!}
∴
J
n
=
−
cos
a
x
(
n
−
1
)
x
n
−
1
−
a
(
n
−
1
)
(
n
−
2
)
(
−
sen
a
x
x
n
−
2
+
a
J
n
−
2
)
{\displaystyle \therefore J_{n}=-{\frac {\cos {ax}}{(n-1)x^{n-1}}}-{\frac {a}{(n-1)(n-2)}}\left(-{\frac {\operatorname {sen} {ax}}{x^{n-2}}}+aJ_{n-2}\right)\,\!}
I
n
=
∫
sen
n
a
x
d
x
{\displaystyle I_{n}=\int \operatorname {sen} ^{n}{ax}\,{\text{d}}x\,\!}
a
n
I
n
=
−
sen
n
−
1
a
x
cos
a
x
+
a
(
n
−
1
)
I
n
−
2
{\displaystyle anI_{n}=-\operatorname {sen} ^{n-1}{ax}\cos {ax}+a(n-1)I_{n-2}\,\!}
J
n
=
∫
cos
n
a
x
d
x
{\displaystyle J_{n}=\int \cos ^{n}{ax}\,{\text{d}}x\,\!}
a
n
J
n
=
sen
a
x
cos
n
−
1
a
x
+
a
(
n
−
1
)
J
n
−
2
{\displaystyle anJ_{n}=\operatorname {sen} {ax}\cos ^{n-1}{ax}+a(n-1)J_{n-2}\,\!}
I
n
=
∫
d
x
sen
n
a
x
{\displaystyle I_{n}=\int {\frac {{\text{d}}x}{\operatorname {sen} ^{n}{ax}}}\,\!}
(
n
−
1
)
I
n
=
−
cos
a
x
a
sen
n
−
1
a
x
+
(
n
−
2
)
I
n
−
2
{\displaystyle (n-1)I_{n}=-{\frac {\cos {ax}}{a\operatorname {sen} ^{n-1}{ax}}}+(n-2)I_{n-2}\,\!}
J
n
=
∫
d
x
cos
n
a
x
{\displaystyle J_{n}=\int {\frac {{\text{d}}x}{\cos ^{n}{ax}}}\,\!}
(
n
−
1
)
J
n
=
sen
a
x
a
cos
n
−
1
a
x
+
(
n
−
2
)
J
n
−
2
{\displaystyle (n-1)J_{n}={\frac {\operatorname {sen} {ax}}{a\cos ^{n-1}{ax}}}+(n-2)J_{n-2}\,\!}
Integral
Fórmula de reducción
I
m
,
n
=
∫
sen
m
a
x
cos
n
a
x
d
x
{\displaystyle I_{m,n}=\int \operatorname {sen} ^{m}{ax}\cos ^{n}{ax}\,{\text{d}}x\,\!}
I
m
,
n
=
{
−
sen
m
−
1
a
x
cos
n
+
1
a
x
a
(
m
+
n
)
+
m
−
1
m
+
n
I
m
−
2
,
n
sen
m
+
1
a
x
cos
n
−
1
a
x
a
(
m
+
n
)
+
n
−
1
m
+
n
I
m
,
n
−
2
{\displaystyle I_{m,n}={\begin{cases}-{\frac {\operatorname {sen} ^{m-1}{ax}\cos ^{n+1}{ax}}{a(m+n)}}+{\frac {m-1}{m+n}}I_{m-2,n}\\{\frac {\operatorname {sen} ^{m+1}{ax}\cos ^{n-1}{ax}}{a(m+n)}}+{\frac {n-1}{m+n}}I_{m,n-2}\\\end{cases}}\,\!}
I
m
,
n
=
∫
d
x
sin
m
a
x
cos
n
a
x
{\displaystyle I_{m,n}=\int {\frac {{\text{d}}x}{\sin ^{m}{ax}\cos ^{n}{ax}}}\,\!}
I
m
,
n
=
{
1
a
(
n
−
1
)
sen
m
−
1
a
x
cos
n
−
1
a
x
+
m
+
n
−
2
n
−
1
I
m
,
n
−
2
−
1
a
(
m
−
1
)
sen
m
−
1
a
x
cos
n
−
1
a
x
+
m
+
n
−
2
m
−
1
I
m
−
2
,
n
{\displaystyle I_{m,n}={\begin{cases}{\frac {1}{a(n-1)\operatorname {sen} ^{m-1}{ax}\cos ^{n-1}{ax}}}+{\frac {m+n-2}{n-1}}I_{m,n-2}\\-{\frac {1}{a(m-1)\operatorname {sen} ^{m-1}{ax}\cos ^{n-1}{ax}}}+{\frac {m+n-2}{m-1}}I_{m-2,n}\\\end{cases}}\,\!}
I
m
,
n
=
∫
sen
m
a
x
cos
n
a
x
d
x
{\displaystyle I_{m,n}=\int {\frac {\operatorname {sen} ^{m}{ax}}{\cos ^{n}{ax}}}\,{\text{d}}x\,\!}
I
m
,
n
=
{
sen
m
−
1
a
x
a
(
n
−
1
)
cos
n
−
1
a
x
−
m
−
1
n
−
1
I
m
−
2
,
n
−
2
sen
m
+
1
a
x
a
(
n
−
1
)
cos
n
−
1
a
x
−
m
−
n
+
2
n
−
1
I
m
,
n
−
2
−
sen
m
−
1
a
x
a
(
m
−
n
)
cos
n
−
1
a
x
+
m
−
1
m
−
n
I
m
−
2
,
n
{\displaystyle I_{m,n}={\begin{cases}{\frac {\operatorname {sen} ^{m-1}{ax}}{a(n-1)\cos ^{n-1}{ax}}}-{\frac {m-1}{n-1}}I_{m-2,n-2}\\{\frac {\operatorname {sen} ^{m+1}{ax}}{a(n-1)\cos ^{n-1}{ax}}}-{\frac {m-n+2}{n-1}}I_{m,n-2}\\-{\frac {\operatorname {sen} ^{m-1}{ax}}{a(m-n)\cos ^{n-1}{ax}}}+{\frac {m-1}{m-n}}I_{m-2,n}\\\end{cases}}\,\!}
I
m
,
n
=
∫
cos
m
a
x
sen
n
a
x
d
x
{\displaystyle I_{m,n}=\int {\frac {\cos ^{m}{ax}}{\operatorname {sen} ^{n}{ax}}}\,{\text{d}}x\,\!}
I
m
,
n
=
{
−
cos
m
−
1
a
x
a
(
n
−
1
)
sen
n
−
1
a
x
−
m
−
1
n
−
1
I
m
−
2
,
n
−
2
−
cos
m
+
1
a
x
a
(
n
−
1
)
sen
n
−
1
a
x
−
m
−
n
+
2
n
−
1
I
m
,
n
−
2
cos
m
−
1
a
x
a
(
m
−
n
)
sen
n
−
1
a
x
+
m
−
1
m
−
n
I
m
−
2
,
n
{\displaystyle I_{m,n}={\begin{cases}-{\frac {\cos ^{m-1}{ax}}{a(n-1)\operatorname {sen} ^{n-1}{ax}}}-{\frac {m-1}{n-1}}I_{m-2,n-2}\\-{\frac {\cos ^{m+1}{ax}}{a(n-1)\operatorname {sen} ^{n-1}{ax}}}-{\frac {m-n+2}{n-1}}I_{m,n-2}\\{\frac {\cos ^{m-1}{ax}}{a(m-n)\operatorname {sen} ^{n-1}{ax}}}+{\frac {m-1}{m-n}}I_{m-2,n}\\\end{cases}}\,\!}
Integral
Fórmula de reducción
I
n
=
∫
x
n
e
a
x
d
x
{\displaystyle I_{n}=\int x^{n}e^{ax}\,{\text{d}}x\,\!}
n
>
0
{\displaystyle n>0\,\!}
I
n
=
x
n
e
a
x
a
−
n
a
I
n
−
1
{\displaystyle I_{n}={\frac {x^{n}e^{ax}}{a}}-{\frac {n}{a}}I_{n-1}\,\!}
I
n
=
∫
x
−
n
e
a
x
d
x
{\displaystyle I_{n}=\int x^{-n}e^{ax}\,{\text{d}}x\,\!}
n
>
0
{\displaystyle n>0\,\!}
n
≠
1
{\displaystyle n\neq 1\,\!}
I
n
=
−
e
a
x
(
n
−
1
)
x
n
−
1
+
a
n
−
1
I
n
−
1
{\displaystyle I_{n}={\frac {-e^{ax}}{(n-1)x^{n-1}}}+{\frac {a}{n-1}}I_{n-1}\,\!}
I
n
=
∫
e
a
x
sen
n
b
x
d
x
{\displaystyle I_{n}=\int e^{ax}\operatorname {sen} ^{n}{bx}\,{\text{d}}x\,\!}
I
n
=
e
a
x
sen
n
−
1
b
x
a
2
+
(
b
n
)
2
(
a
sen
b
x
−
b
n
cos
b
x
)
+
n
(
n
−
1
)
b
2
a
2
+
(
b
n
)
2
I
n
−
2
{\displaystyle I_{n}={\frac {e^{ax}\operatorname {sen} ^{n-1}{bx}}{a^{2}+(bn)^{2}}}\left(a\operatorname {sen} bx-bn\cos bx\right)+{\frac {n(n-1)b^{2}}{a^{2}+(bn)^{2}}}I_{n-2}\,\!}
I
n
=
∫
e
a
x
cos
n
b
x
d
x
{\displaystyle I_{n}=\int e^{ax}\cos ^{n}{bx}\,{\text{d}}x\,\!}
I
n
=
e
a
x
cos
n
−
1
b
x
a
2
+
(
b
n
)
2
(
a
cos
b
x
+
b
n
sen
b
x
)
+
n
(
n
−
1
)
b
2
a
2
+
(
b
n
)
2
I
n
−
2
{\displaystyle I_{n}={\frac {e^{ax}\cos ^{n-1}{bx}}{a^{2}+(bn)^{2}}}\left(a\cos bx+bn\operatorname {sen} bx\right)+{\frac {n(n-1)b^{2}}{a^{2}+(bn)^{2}}}I_{n-2}\,\!}
Anton, Bivens, Davis, Cálculo, 7.ª edición. 
![{\displaystyle I_{5}=\int \cos ^{5}(x)\;dx={\frac {1}{5}}\cos ^{4}(x)\operatorname {sen} x+{\frac {4}{5}}\left[{\frac {1}{3}}\cos ^{2}(x)\operatorname {sen} x+{\frac {2}{3}}\operatorname {sen} x\right]+C,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ec92d6c5c81813f3af15348b5037366605bcb7f)
![{\displaystyle I_{m,n}={\begin{cases}-{\frac {1}{(n-1)(bp-aq)}}\left[{\frac {1}{(ax+b)^{m-1}(px+q)^{n-1}}}+a(m+n-2)I_{m,n-1}\right]\\{\frac {1}{(m-1)(bp-aq)}}\left[{\frac {1}{(ax+b)^{m-1}(px+q)^{n-1}}}+p(m+n-2)I_{m-1,n}\right]\end{cases}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22e62922721540ee6636beadc5183d305fa0a98b)
![{\displaystyle I_{m,n}={\begin{cases}-{\frac {1}{(n-1)(bp-aq)}}\left[{\frac {(ax+b)^{m+1}}{(px+q)^{n-1}}}+a(n-m-2)I_{m,n-1}\right]\\-{\frac {1}{(n-m-1)p}}\left[{\frac {(ax+b)^{m}}{(px+q)^{n-1}}}+m(bp-aq)I_{m-1,n}\right]\\-{\frac {1}{(n-1)p}}\left[{\frac {(ax+b)^{m}}{(px+q)^{n-1}}}-amI_{m-1,n-1}\right]\end{cases}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18cf63aea753eb3517e589623b4728e1545c012b)
![{\displaystyle I_{n}=-{\frac {\operatorname {sen} {ax}}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}\left[{\frac {\cos {ax}}{(n-2)x^{n-2}}}+{\frac {a}{n-2}}I_{n-2}\right]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05a7ffd6ccce99a41701877d1c01402e8f210595)
![{\displaystyle J_{n}=-{\frac {\cos {ax}}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}\left[-{\frac {\operatorname {sen} {ax}}{(n-2)x^{n-2}}}+{\frac {a}{n-2}}J_{n-2}\right]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee33fbb14f8651180e6eb9a542e78935f33f477c)
