漸化式による積分
漸化式の表 > 有理関数

以下の積分は、これらを含む。
線形根号の因子  {\sqrt {ax+b}}\,\!
線形因子  {px+q}\,\! と線形根号  {\sqrt {ax+b}}\,\!
二次因子  x^{2}+a^{2}\,\!
二次因子  x^{2}-a^{2}\,\!, for  x>a\,\!
二次因子  a^{2}-x^{2}\,\!, for  x<a\,\!
(既約) 二次因子  ax^{2}+bx+c\,\!
既約多項式因子の根号  {\sqrt {ax^{2}+bx+c}}\,\!
| 積分 | 漸化式 |
| --- | --- |
 I_{n}=\int {\frac {x^{n}}{\sqrt {ax+b}}}\,{\text{d}}x\,\!:  I_{n}={\frac {2x^{n}{\sqrt {ax+b}}}{a(2n+1)}}-{\frac {2nb}{a(2n+1)}}I_{n-1}\,\!
 I_{n}=\int {\frac {{\text{d}}x}{x^{n}{\sqrt {ax+b}}}}\,\!:  I_{n}=-{\frac {\sqrt {ax+b}}{(n-1)bx^{n-1}}}-{\frac {a(2n-3)}{2b(n-1)}}I_{n-1}\,\!
 I_{n}=\int x^{n}{\sqrt {ax+b}}\,{\text{d}}x\,\!:  I_{n}={\frac {2x^{n}{\sqrt {(ax+b)^{3}}}}{a(2n+3)}}-{\frac {2nb}{a(2n+3)}}I_{n-1}\,\!
 I_{m,n}=\int {\frac {{\text{d}}x}{(ax+b)^{m}(px+q)^{n}}}\,\!:  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}=\int {\frac {(ax+b)^{m}}{(px+q)^{n}}}\,{\text{d}}x\,\!:  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}}\,\!
| 積分 | 漸化式 |
| --- | --- |
 I_{n}=\int {\frac {(px+q)^{n}}{\sqrt {ax+b}}}\,{\text{d}}x\,\!:  \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}={\frac {2(px+q)^{n}{\sqrt {ax+b}}}{a(2n+1)}}+{\frac {2n(aq-bp)}{a(2n+1)}}I_{n-1}\,\!
 I_{n}=\int {\frac {{\text{d}}x}{(px+q)^{n}{\sqrt {ax+b}}}}\,\!:  \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}=-{\frac {\sqrt {ax+b}}{(n-1)(aq-bp)(px+q)^{n-1}}}+{\frac {a(2n-3)}{2(n-1)(aq-bp)}}I_{n-1}\,\!
| 積分 | 漸化式 |
| --- | --- |
 I_{n}=\int {\frac {{\text{d}}x}{(x^{2}+a^{2})^{n}}}\,\!:  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}=\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}\,\!
 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}\,\!
| 積分 | 漸化式 |
| --- | --- |
 I_{n}=\int {\frac {{\text{d}}x}{(x^{2}-a^{2})^{n}}}\,\!:  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}=\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}\,\!
 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}\,\!
| 積分 | 漸化式 |
| --- | --- |
 I_{n}=\int {\frac {{\text{d}}x}{(a^{2}-x^{2})^{n}}}\,\!:  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}=\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}\,\!
 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}\,\!
| 積分 | 漸化式 |
| --- | --- |
 I_{n}=\int {\frac {{\text{d}}x}{{x^{n}}(ax^{2}+bx+c)}}\,\!:  -cI_{n}={\frac {1}{x^{n-1}(n-1)}}+bI_{n-1}+aI_{n-2}\,\!
 I_{m,n}=\int {\frac {x^{m}\,{\text{d}}x}{(ax^{2}+bx+c)^{n}}}\,\!:  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}=\int {\frac {{\text{d}}x}{x^{m}(ax^{2}+bx+c)^{n}}}\,\!:  -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}\,\!

| 積分 | 漸化式 |
| --- | --- |
 I_{n}=\int (ax^{2}+bx+c)^{n}\,{\text{d}}x\,\!:  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}=\int {\frac {1}{(ax^{2}+bx+c)^{n}}}\,{\text{d}}x\,\!:  (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}}}\,\!