적분 미적분학 에서 적분의 점화식 은 점화식 의 형태로 된 적분 공식이다. 주로 초등함수 의 거듭제곱, 초월함수 와 n차다항식의 곱의 형태 같은 정수 매개변수 를 포함하는 수식 을 직접 적분하기 어려울 때 쓰인다.
이러한 점화식과 다른 적분법 을 이용하면 피적분함수를 정수 매개변수의 값이 더 작은 동일하거나 유사한 형태의 식으로 바꾸어 쉽게 적분이 가능하도록 단순화 할 수 있다.[ 1]
적분의 점화식은 치환 적분 , 부분 적분 , 삼각 치환 , 부분 분수 에 의한 적분 등과 같은 일반적인 적분법을 이용하여 유도할 수 있다. 핵심은 정수 매개 변수를 이용하여 나타낼 수 있는 함수
I
n
{\displaystyle I_{n}
I
n
−
1
{\displaystyle I_{n-1}
I
n
−
2
{\displaystyle I_{n-2}
I
n
=
∫
f
(
x
,
n
)
d
x
,
{\displaystyle I_{n}=\int f(x,n)\,{\text{d}x,}
을
k
<
n
{\displaystyle k<n}
k
{\displaystyle k}
I
k
=
∫
f
(
x
,
k
)
d
x
,
{\displaystyle I_{k}=\int f(x,k)\,{\text{d}x,}
의 형태로 나타내는 것이다.
점화식을 이용하여 함수를 적분하기 위해서
I
n
{\displaystyle I_{n}
I
n
−
1
{\displaystyle I_{n-1}
I
n
−
2
{\displaystyle I_{n-2}
I
n
{\displaystyle I_{n}
I
n
{\displaystyle I_{n}
[ 2]
아래는 점화식을 이용한 적분 과정의 예시이다.
코사인함수의 적분
일반적으로
∫
cos
n
x
d
x
,
{\displaystyle \int \cos ^{n}x\,{\text{d}x,\,\!}
와 같은 적분은 적분의 점화식을 이용하여 적분할 수 있다.
n = 1, 2 ... 30인 경우의
∫
cos
n
(
x
)
d
x
{\displaystyle \int \cos ^{n}(x)\,{\text{d}x\!}
I
n
=
∫
cos
n
x
d
x
.
{\displaystyle I_{n}=\int \cos ^{n}x\,{\text{d}x.\,\!}
라고 하자.
I
n
=
∫
cos
n
−
1
x
cos
x
d
x
,
{\displaystyle I_{n}=\int \cos ^{n-1}x\cos x\,{\text{d}x,\,\!}
cos
x
d
x
=
d
(
sin
x
)
,
{\displaystyle \cos x\,{\text{d}x={\text{d}(\sin x),\,\!}
라고 치환하면
I
n
=
∫
cos
n
−
1
x
d
(
sin
x
)
.
{\displaystyle I_{n}=\int \cos ^{n-1}x\,{\text{d}(\sin x).\!}
부분 적분을 이용하면
∫
cos
n
x
d
x
=
cos
n
−
1
x
sin
x
−
∫
sin
x
d
(
cos
n
−
1
x
)
=
cos
n
−
1
x
sin
x
+
(
n
−
1
)
∫
sin
x
cos
n
−
2
x
sin
x
d
x
=
cos
n
−
1
x
sin
x
+
(
n
−
1
)
∫
cos
n
−
2
x
sin
2
x
d
x
=
cos
n
−
1
x
sin
x
+
(
n
−
1
)
∫
cos
n
−
2
x
(
1
−
cos
2
x
)
d
x
=
cos
n
−
1
x
sin
x
+
(
n
−
1
)
∫
cos
n
−
2
x
d
x
−
(
n
−
1
)
∫
cos
n
x
d
x
=
cos
n
−
1
x
sin
x
+
(
n
−
1
)
I
n
−
2
−
(
n
−
1
)
I
n
,
{\displaystyle {\begin{aligned}\int \cos ^{n}x\,{\text{d}x&=\cos ^{n-1}x\sin x-\int \sin x\,{\text{d}(\cos ^{n-1}x)\\&=\cos ^{n-1}x\sin x+(n-1)\int \sin x\cos ^{n-2}x\sin x\,{\text{d}x\\&=\cos ^{n-1}x\sin x+(n-1)\int \cos ^{n-2}x\sin ^{2}x\,{\text{d}x\\&=\cos ^{n-1}x\sin x+(n-1)\int \cos ^{n-2}x(1-\cos ^{2}x)\,{\text{d}x\\&=\cos ^{n-1}x\sin x+(n-1)\int \cos ^{n-2}x\,{\text{d}x-(n-1)\int \cos ^{n}x\,{\text{d}x\\&=\cos ^{n-1}x\sin x+(n-1)I_{n-2}-(n-1)I_{n},\end{aligned}\,}
I
n
+
(
n
−
1
)
I
n
=
cos
n
−
1
x
sin
x
+
(
n
−
1
)
I
n
−
2
,
{\displaystyle I_{n}\ +(n-1)I_{n}\ =\cos ^{n-1}x\sin x\ +\ (n-1)I_{n-2},\,}
n
I
n
=
cos
n
−
1
(
x
)
sin
x
+
(
n
−
1
)
I
n
−
2
,
{\displaystyle nI_{n}\ =\cos ^{n-1}(x)\sin x\ +(n-1)I_{n-2},\,}
I
n
=
1
n
cos
n
−
1
x
sin
x
+
n
−
1
n
I
n
−
2
,
{\displaystyle I_{n}\ ={\frac {1}{n}\cos ^{n-1}x\sin x\ +{\frac {n-1}{n}I_{n-2},\,}
따라서 적분의 점화식은 다음과 같다.
∫
cos
n
x
d
x
=
1
n
cos
n
−
1
x
sin
x
+
n
−
1
n
∫
cos
n
−
2
x
d
x
.
{\displaystyle \int \cos ^{n}x\,{\text{d}x\ ={\frac {1}{n}\cos ^{n-1}x\sin x+{\frac {n-1}{n}\int \cos ^{n-2}x\,{\text{d}x.\!}
이 예시를 보충하기 위해 n = 5를 대입해 보면.
I
5
=
∫
cos
5
x
d
x
.
{\displaystyle I_{5}=\int \cos ^{5}x\,{\text{d}x.\,\!}
n=5, n=3을 대입하면
n
=
5
,
I
5
=
1
5
cos
4
x
sin
x
+
4
5
I
3
,
{\displaystyle n=5,\quad I_{5}={\tfrac {1}{5}\cos ^{4}x\sin x+{\tfrac {4}{5}I_{3},\,}
n
=
3
,
I
3
=
1
3
cos
2
x
sin
x
+
2
3
I
1
,
{\displaystyle n=3,\quad I_{3}={\tfrac {1}{3}\cos ^{2}x\sin x+{\tfrac {2}{3}I_{1},\,}
결과를 대입하면
∵
I
1
=
∫
cos
x
d
x
=
sin
x
+
C
1
,
{\displaystyle \because I_{1}\ =\int \cos x\,{\text{d}x=\sin x+C_{1},\,}
∴
I
3
=
1
3
cos
2
x
sin
x
+
2
3
sin
x
+
C
2
,
C
2
=
2
3
C
1
,
{\displaystyle \therefore I_{3}\ ={\tfrac {1}{3}\cos ^{2}x\sin x+{\tfrac {2}{3}\sin x+C_{2},\quad C_{2}\ ={\tfrac {2}{3}C_{1},\,}
I
5
=
1
5
cos
4
x
sin
x
+
4
5
[
1
3
cos
2
x
sin
x
+
2
3
sin
x
]
+
C
,
{\displaystyle I_{5}\ ={\frac {1}{5}\cos ^{4}x\sin x+{\frac {4}{5}\left[{\frac {1}{3}\cos ^{2}x\sin x+{\frac {2}{3}\sin x\right]+C,\,}
여기서 C 는 적분상수이다.
지수함수의 적분
또 다른 전형적인 예는 다음과 같다.
∫
x
n
e
a
x
d
x
.
{\displaystyle \int x^{n}e^{ax}\,{\text{d}x.\,\!}
I
n
=
∫
x
n
e
a
x
d
x
.
{\displaystyle I_{n}=\int x^{n}e^{ax}\,{\text{d}x.\,\!}
라고 하자.
치환 적분을 하면
x
n
d
x
=
d
(
x
n
+
1
)
n
+
1
,
{\displaystyle x^{n}\,{\text{d}x={\frac {\text{d}(x^{n+1})}{n+1},\,\!}
로 치환하면
I
n
=
1
n
+
1
∫
e
a
x
d
(
x
n
+
1
)
,
{\displaystyle I_{n}={\frac {1}{n+1}\int e^{ax}\,{\text{d}(x^{n+1}),\!}
부분 적분을 하면
∫
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}\,{\text{d}(x^{n+1})&=x^{n+1}e^{ax}-\int x^{n+1}\,{\text{d}(e^{ax})\\&=x^{n+1}e^{ax}-a\int x^{n+1}e^{ax}\,{\text{d}x,\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},\!}
n + 1 → n , n → n – 1으로 바꾸면
n
I
n
−
1
=
x
n
e
a
x
−
a
I
n
,
{\displaystyle nI_{n-1}=x^{n}e^{ax}-aI_{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),\,\!}
따라서 적분의 점화식은 다음과 같다.
∫
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}\,{\text{d}x={\frac {1}{a}\left(x^{n}e^{ax}-n\int x^{n-1}e^{ax}\,{\text{d}x\right).\!}
e
a
x
{\displaystyle e^{ax}
e
a
x
d
x
=
d
(
e
a
x
)
a
,
{\displaystyle e^{ax}\,{\text{d}x={\frac {\text{d}(e^{ax})}{a},\,\!}
I
n
=
1
a
∫
x
n
d
(
e
a
x
)
,
{\displaystyle I_{n}={\frac {1}{a}\int x^{n}\,{\text{d}(e^{ax}),\!}
∫
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}\,{\text{d}(e^{ax})&=x^{n}e^{ax}-\int e^{ax}\,{\text{d}(x^{n})\\&=x^{n}e^{ax}-n\int e^{ax}x^{n-1}\,{\text{d}x,\end{aligned}\!}
결과를 다시 대입하면 점화식을 구할 수 있다.
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),\,\!}
즉
∫
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}\,{\text{d}x={\frac {1}{a}\left(x^{n}e^{ax}-n\int x^{n-1}e^{ax}\,{\text{d}x\right).\!}
다음의 적분[ 3]
선형 라디칼
a
x
+
b
{\displaystyle {\sqrt {ax+b}\,\!}
선형 인수
p
x
+
q
{\displaystyle {px+q}\,\!}
a
x
+
b
{\displaystyle {\sqrt {ax+b}\,\!}
이차 식
x
2
+
a
2
{\displaystyle x^{2}+a^{2}\,\!}
이차식
x
2
−
a
2
{\displaystyle x^{2}-a^{2}\,\!}
x
>
a
{\displaystyle x>a\,\!}
이차식
a
2
−
x
2
{\displaystyle a^{2}-x^{2}\,\!}
x
<
a
{\displaystyle x<a\,\!}
(기약 다항식 )
a
x
2
+
b
x
+
c
{\displaystyle ax^{2}+bx+c\,\!}
기약 다항식
a
x
2
+
b
x
+
c
{\displaystyle {\sqrt {ax^{2}+bx+c}\,\!}
적분
적분의 점화식
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}\,\!}
적분
적분의 점화식
I
n
=
∫
(
p
x
+
q
)
n
a
x
+
b
d
x
{\displaystyle I_{n}=\int {\frac {(px+q)^{n}{\sqrt {ax+b}\,{\text{d}x\,\!}
∫
(
p
x
+
q
)
n
a
x
+
b
d
x
=
2
(
p
x
+
q
)
n
+
1
a
x
+
b
p
(
2
n
+
3
)
+
b
p
−
a
q
p
(
2
n
+
3
)
I
n
{\displaystyle \int (px+q)^{n}{\sqrt {ax+b}\,{\text{d}x={\frac {2(px+q)^{n+1}{\sqrt {ax+b}{p(2n+3)}+{\frac {bp-aq}{p(2n+3)}I_{n}\,\!}
I
n
=
2
(
p
x
+
q
)
n
a
x
+
b
a
(
2
n
+
1
)
+
2
n
(
a
q
−
b
p
)
a
(
2
n
+
1
)
I
n
−
1
{\displaystyle I_{n}={\frac {2(px+q)^{n}{\sqrt {ax+b}{a(2n+1)}+{\frac {2n(aq-bp)}{a(2n+1)}I_{n-1}\,\!}
I
n
=
∫
d
x
(
p
x
+
q
)
n
a
x
+
b
{\displaystyle I_{n}=\int {\frac {\text{d}x}{(px+q)^{n}{\sqrt {ax+b}\,\!}
∫
a
x
+
b
(
p
x
+
q
)
n
d
x
=
−
a
x
+
b
p
(
n
−
1
)
(
p
x
+
q
)
n
−
1
+
a
2
p
(
n
−
1
)
I
n
{\displaystyle \int {\frac {\sqrt {ax+b}{(px+q)^{n}\,{\text{d}x=-{\frac {\sqrt {ax+b}{p(n-1)(px+q)^{n-1}+{\frac {a}{2p(n-1)}I_{n}\,\!}
I
n
=
−
a
x
+
b
(
n
−
1
)
(
a
q
−
b
p
)
(
p
x
+
q
)
n
−
1
+
a
(
2
n
−
3
)
2
(
n
−
1
)
(
a
q
−
b
p
)
I
n
−
1
{\displaystyle I_{n}=-{\frac {\sqrt {ax+b}{(n-1)(aq-bp)(px+q)^{n-1}+{\frac {a(2n-3)}{2(n-1)(aq-bp)}I_{n-1}\,\!}
적분
적분의 점화식
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}\,\!}
적분
적분의 점화식
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}\,\!}
적분
적분의 점화식
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}\,\!}
적분
적분의 점화식
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}\,\!}
적분
적분의 점화식
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}\,\!}
지수법칙 에 따라:
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\,\!}
다음의 적분[ 4]
사인
코사인
사인 및 코사인의 곱 또는 나누었을 때의 몫
x 의 거듭 제곱과 지수함수의 곱사인 / 코사인과 지수함수의 곱
적분
적분의 점화식
I
n
=
∫
x
n
sin
a
x
d
x
{\displaystyle I_{n}=\int x^{n}\sin {ax}\,{\text{d}x\,\!}
a
2
I
n
=
−
a
x
n
cos
a
x
+
n
x
n
−
1
sin
a
x
−
n
(
n
−
1
)
I
n
−
2
{\displaystyle a^{2}I_{n}=-ax^{n}\cos {ax}+nx^{n-1}\sin {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
sin
a
x
+
n
x
n
−
1
cos
a
x
−
n
(
n
−
1
)
J
n
−
2
{\displaystyle a^{2}J_{n}=ax^{n}\sin {ax}+nx^{n-1}\cos {ax}-n(n-1)J_{n-2}\,\!}
I
n
=
∫
sin
a
x
x
n
d
x
{\displaystyle I_{n}=\int {\frac {\sin {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
=
−
sin
a
x
(
n
−
1
)
x
n
−
1
+
a
n
−
1
J
n
−
1
{\displaystyle I_{n}=-{\frac {\sin {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}\,\!}
공식을 결합하여 I n 의 개별적인 방정식을 얻을 수 있다.
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
=
−
sin
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 {\sin {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
=
−
sin
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 {\sin {ax}{(n-1)x^{n-1}-{\frac {a}{(n-1)(n-2)}\left({\frac {\cos {ax}{x^{n-2}+aI_{n-2}\right)\,\!}
그리고 J n :
I
n
−
1
=
−
sin
a
x
(
n
−
2
)
x
n
−
2
+
a
n
−
2
J
n
−
2
{\displaystyle I_{n-1}=-{\frac {\sin {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
[
−
sin
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 {\sin {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
)
(
−
sin
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 {\sin {ax}{x^{n-2}+aJ_{n-2}\right)\,\!}
I
n
=
∫
sin
n
a
x
d
x
{\displaystyle I_{n}=\int \sin ^{n}{ax}\,{\text{d}x\,\!}
a
n
I
n
=
−
sin
n
−
1
a
x
cos
a
x
+
a
(
n
−
1
)
I
n
−
2
{\displaystyle anI_{n}=-\sin ^{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
=
sin
a
x
cos
n
−
1
a
x
+
a
(
n
−
1
)
J
n
−
2
{\displaystyle anJ_{n}=\sin {ax}\cos ^{n-1}{ax}+a(n-1)J_{n-2}\,\!}
I
n
=
∫
d
x
sin
n
a
x
{\displaystyle I_{n}=\int {\frac {\text{d}x}{\sin ^{n}{ax}\,\!}
(
n
−
1
)
I
n
=
−
cos
a
x
a
sin
n
−
1
a
x
+
(
n
−
2
)
I
n
−
2
{\displaystyle (n-1)I_{n}=-{\frac {\cos {ax}{a\sin ^{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
=
sin
a
x
a
cos
n
−
1
a
x
+
(
n
−
2
)
J
n
−
2
{\displaystyle (n-1)J_{n}={\frac {\sin {ax}{a\cos ^{n-1}{ax}+(n-2)J_{n-2}\,\!}
적분
적분의 점화식
I
m
,
n
=
∫
sin
m
a
x
cos
n
a
x
d
x
{\displaystyle I_{m,n}=\int \sin ^{m}{ax}\cos ^{n}{ax}\,{\text{d}x\,\!}
I
m
,
n
=
{
−
sin
m
−
1
a
x
cos
n
+
1
a
x
a
(
m
+
n
)
+
m
−
1
m
+
n
I
m
−
2
,
n
sin
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 {\sin ^{m-1}{ax}\cos ^{n+1}{ax}{a(m+n)}+{\frac {m-1}{m+n}I_{m-2,n}\\{\frac {\sin ^{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
)
sin
m
−
1
a
x
cos
n
−
1
a
x
+
m
+
n
−
2
n
−
1
I
m
,
n
−
2
−
1
a
(
m
−
1
)
sin
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)\sin ^{m-1}{ax}\cos ^{n-1}{ax}+{\frac {m+n-2}{n-1}I_{m,n-2}\\-{\frac {1}{a(m-1)\sin ^{m-1}{ax}\cos ^{n-1}{ax}+{\frac {m+n-2}{m-1}I_{m-2,n}\\\end{cases}\,\!}
I
m
,
n
=
∫
sin
m
a
x
cos
n
a
x
d
x
{\displaystyle I_{m,n}=\int {\frac {\sin ^{m}{ax}{\cos ^{n}{ax}\,{\text{d}x\,\!}
I
m
,
n
=
{
sin
m
−
1
a
x
a
(
n
−
1
)
cos
n
−
1
a
x
−
m
−
1
n
−
1
I
m
−
2
,
n
−
2
sin
m
+
1
a
x
a
(
n
−
1
)
cos
n
−
1
a
x
−
m
−
n
+
2
n
−
1
I
m
,
n
−
2
−
sin
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 {\sin ^{m-1}{ax}{a(n-1)\cos ^{n-1}{ax}-{\frac {m-1}{n-1}I_{m-2,n-2}\\{\frac {\sin ^{m+1}{ax}{a(n-1)\cos ^{n-1}{ax}-{\frac {m-n+2}{n-1}I_{m,n-2}\\-{\frac {\sin ^{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
sin
n
a
x
d
x
{\displaystyle I_{m,n}=\int {\frac {\cos ^{m}{ax}{\sin ^{n}{ax}\,{\text{d}x\,\!}
I
m
,
n
=
{
−
cos
m
−
1
a
x
a
(
n
−
1
)
sin
n
−
1
a
x
−
m
−
1
n
−
1
I
m
−
2
,
n
−
2
−
cos
m
+
1
a
x
a
(
n
−
1
)
sin
n
−
1
a
x
−
m
−
n
+
2
n
−
1
I
m
,
n
−
2
cos
m
−
1
a
x
a
(
m
−
n
)
sin
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)\sin ^{n-1}{ax}-{\frac {m-1}{n-1}I_{m-2,n-2}\\-{\frac {\cos ^{m+1}{ax}{a(n-1)\sin ^{n-1}{ax}-{\frac {m-n+2}{n-1}I_{m,n-2}\\{\frac {\cos ^{m-1}{ax}{a(m-n)\sin ^{n-1}{ax}+{\frac {m-1}{m-n}I_{m-2,n}\\\end{cases}\,\!}
적분
적분의 점화식
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
sin
n
b
x
d
x
{\displaystyle I_{n}=\int e^{ax}\sin ^{n}{bx}\,{\text{d}x\,\!}
I
n
=
e
a
x
sin
n
−
1
b
x
a
2
+
(
b
n
)
2
(
a
sin
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}\sin ^{n-1}{bx}{a^{2}+(bn)^{2}\left(a\sin 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
sin
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\sin bx\right)+{\frac {n(n-1)b^{2}{a^{2}+(bn)^{2}I_{n-2}\,\!}
↑ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3
↑ Further Elementary Analysis, R.I. Porter, G. Bell & Sons Ltd, 1978, ISBN 0-7135-1594-5
↑ http://www.sosmath.com/tables/tables.html -> Indefinite integrals list↑ http://www.sosmath.com/tables/tables.html -> Indefinite integrals list
Anton, Bivens, Davis, Calculus, 7th edition. 
![{\displaystyle I_{5}\ ={\frac {1}{5}\cos ^{4}x\sin x+{\frac {4}{5}\left[{\frac {1}{3}\cos ^{2}x\sin x+{\frac {2}{3}\sin x\right]+C,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7803bf2d86071f40537d4590e76ab498354b7c7f)
![{\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 {\sin {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/c59ab79c10f4eed2620c496fac8b476174a84477)
![{\displaystyle J_{n}=-{\frac {\cos {ax}{(n-1)x^{n-1}-{\frac {a}{n-1}\left[-{\frac {\sin {ax}{(n-2)x^{n-2}+{\frac {a}{n-2}J_{n-2}\right]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2ed60b99d5d7b4dfa736c3627c68021c847b44)
