原始関数の一覧

本項は、原始関数の一覧（げんしかんすうのいちらん）である。以下、積分定数は${\displaystyle C}$とする。

a x + b {\displaystyle ax+b} を含む積分

${\displaystyle \int {\frac {1}{ax+b}}\,dx={\frac {1}{a}}\ln \left|ax+b\right|+C}$

${\displaystyle \int {\frac {x}{ax+b}}\,dx={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln \left|ax+b\right|+C}$

${\displaystyle \int {\frac {x^{2}}{ax+b}}\,dx={\frac {1}{2a^{3}}}(a^{2}x^{2}-2abx+2b^{2}\ln \left|ax+b\right|)+C}$

${\displaystyle \int {\frac {1}{x(ax+b)}}\,dx=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|+C}$

${\displaystyle \int {\frac {1}{x^{2}(ax+b)}}\,dx={\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|-{\frac {1}{bx}}+C}$

a + b x {\displaystyle {\sqrt {a+bx}}} を含む積分

${\displaystyle \int x{\sqrt {a+bx}}\,dx={\frac {2}{15b^{2}}}(3bx-2a)(a+bx)^{\frac {3}{2}}+C}$

${\displaystyle \int x^{2}{\sqrt {a+bx}}\,dx={\frac {2}{105b^{3}}}(15b^{2}x^{2}-12abx+8a^{2})(a+bx)^{\frac {3}{2}}+C}$

${\displaystyle \int x^{n}{\sqrt {a+bx}}\,dx={\frac {2}{b(2n+3)}}x^{n}(a+bx)^{\frac {3}{2}}-{\frac {2na}{b(2n+3)}}\int x^{n-1}{\sqrt {a+bx}}dx}$

${\displaystyle \int {\frac {\sqrt {a+bx}}{x}}\,dx=2{\sqrt {a+bx}}+a\int {\frac {1}{x{\sqrt {a+bx}}}}dx}$

${\displaystyle \int {\frac {\sqrt {a+bx}}{x^{n}}}\,dx={\frac {-1}{a(n-1)}}{\frac {(a+bx)^{\frac {3}{2}}}{x^{n-1}}}-{\frac {(2n-5)b}{2a(n-1)}}\int {\frac {\sqrt {a+bx}}{x^{n-1}}}dx,n\neq 1}$

${\displaystyle \int {\frac {1}{x{\sqrt {a+bx}}}}\,dx={\frac {1}{\sqrt {a}}}\ln \left({\frac {{\sqrt {a+bx}}-{\sqrt {a}}}{{\sqrt {a+bx}}+{\sqrt {a}}}}\right)+C,a>0}$

${\displaystyle ={\frac {2}{\sqrt {-a}}}\arctan {\sqrt {\frac {a+bx}{-a}}}+C,a<0}$

${\displaystyle \int {\frac {1}{x^{n}{\sqrt {a+bx}}}}\,dx={\frac {-1}{a(n-1)}}{\frac {\sqrt {a+bx}}{x^{n-1}}}-{\frac {(2n-3)b}{2a(n-1)}}\int {\frac {1}{x^{n-1}}}{\sqrt {a+bx}}dx,n\neq 1}$

x 2 ± α 2 ( α ≠ 0 ) {\displaystyle x^{2}\pm {\alpha }^{2}(\alpha \neq 0)} を含む積分

${\displaystyle \int {\frac {1}{x^{2}+\alpha ^{2}}}\,dx={\frac {1}{\alpha }}\arctan {\frac {x}{\alpha }}+C}$

${\displaystyle \int {\frac {1}{\pm x^{2}\mp \alpha ^{2}}}\,dx={\frac {1}{2\alpha }}\ln \left({\dfrac {x\mp \alpha }{\pm x+\alpha }}\right)+C}$

a x 2 + b {\displaystyle ax^{2}+b} を含む積分

${\displaystyle \int {\frac {1}{ax^{2}+b}}\,dx={\frac {1}{\sqrt {ab}}}\arctan {\sqrt {\frac {a}{b}}}x+C}$

a x 2 + b x + c ( a ≠ 0 ) {\displaystyle ax^{2}+bx+c(a\neq 0)} を含む積分

${\displaystyle \int (ax^{2}+bx+c)\,dx={\frac {ax^{3}}{3}}+{\frac {bx^{2}}{2}}+cx+C}$

a 2 + x 2 ( a &gt; 0 ) {\displaystyle {\sqrt {a^{2}+x^{2}}}\;(a&gt;0)} を含む積分

${\displaystyle \int {\sqrt {a^{2}+x^{2}}}\,dx={\frac {1}{2}}x{\sqrt {a^{2}+x^{2}}}+{\frac {1}{2}}a^{2}\ln \left(x+{\sqrt {a^{2}+x^{2}}}\right)+C}$

${\displaystyle \int x^{2}{\sqrt {a^{2}+x^{2}}}\,dx={\frac {1}{8}}x(a^{2}+2x^{2}){\sqrt {a^{2}+x^{2}}}-{\frac {1}{8}}a^{4}\ln \left(x+{\sqrt {a^{2}+x^{2}}}\right)+C}$

${\displaystyle \int {\frac {\sqrt {a^{2}+x^{2}}}{x}}\,dx={\sqrt {a^{2}+x^{2}}}-a\ln \left({\frac {a+{\sqrt {a^{2}+x^{2}}}}{x}}\right)+C}$

${\displaystyle \int {\frac {\sqrt {a^{2}+x^{2}}}{x^{2}}}\,dx=\ln \left(x+{\sqrt {a^{2}+x^{2}}}\right)-{\frac {\sqrt {a^{2}+x^{2}}}{x}}+C}$

${\displaystyle \int {\frac {1}{\sqrt {a^{2}+x^{2}}}}\,dx=\ln \left(x+{\sqrt {a^{2}+x^{2}}}\right)+C}$

${\displaystyle \int {\frac {x^{2}}{\sqrt {a^{2}+x^{2}}}}\,dx={\frac {1}{2}}x{\sqrt {a^{2}+x^{2}}}-{\frac {1}{2}}a^{2}\ln \left({\sqrt {a^{2}+x^{2}}}+x\right)+C}$

${\displaystyle \int {\frac {1}{x{\sqrt {a^{2}+x^{2}}}}}\,dx={\frac {1}{a}}\ln \left({\frac {x}{a+{\sqrt {a^{2}+x^{2}}}}}\right)+C}$

${\displaystyle \int {\frac {1}{x^{2}{\sqrt {a^{2}+x^{2}}}}}\,dx=-{\frac {\sqrt {a^{2}+x^{2}}}{a^{2}x}}+C}$

x 2 − a 2 ( x 2 &gt; a 2 ) {\displaystyle {\sqrt {x^{2}-a^{2}}}\;(x^{2}&gt;a^{2})} を含む積分

${\displaystyle \int {\frac {1}{\sqrt {x^{2}-a^{2}}}}\,dx=\ln \left(x+{\sqrt {x^{2}-a^{2}}}\right)+C}$

a 2 − x 2 ( a 2 &gt; x 2 ) {\displaystyle {\sqrt {a^{2}-x^{2}}}\;(a^{2}&gt;x^{2})} を含む積分

${\displaystyle \int {\frac {1}{\sqrt {a^{2}-x^{2}}}}\,dx=\arcsin {\frac {x}{a}}+C=-\arccos {\frac {x}{a}}+C}$

${\displaystyle \int {\sqrt {a^{2}-x^{2}}}\,dx={\frac {1}{2}}x{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}+C}$

${\displaystyle \int x^{2}{\sqrt {a^{2}-x^{2}}}\,dx={\frac {1}{8}}x(2x^{2}-a^{2}){\sqrt {a^{2}-x^{2}}}+{\frac {1}{8}}a^{4}\arcsin {\frac {x}{a}}+C}$

${\displaystyle \int {\frac {\sqrt {a^{2}-x^{2}}}{x}}\,dx={\sqrt {a^{2}-x^{2}}}-a\ln \left({\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right)+C}$

${\displaystyle \int {\frac {\sqrt {a^{2}-x^{2}}}{x^{2}}}\,dx=-{\frac {\sqrt {a^{2}-x^{2}}}{x}}-\arcsin {\frac {x}{a}}+C}$

${\displaystyle \int {\frac {1}{x{\sqrt {a^{2}-x^{2}}}}}\,dx=-{\frac {1}{a}}\ln \left({\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right)+C}$

${\displaystyle \int {\frac {x^{2}}{\sqrt {a^{2}-x^{2}}}}\,dx=-{\frac {1}{2}}x{\sqrt {a^{2}-x^{2}}}+{\frac {1}{2}}a^{2}\arcsin {\frac {x}{a}}+C}$

${\displaystyle \int {\frac {1}{x^{2}{\sqrt {a^{2}-x^{2}}}}}\,dx=-{\frac {\sqrt {a^{2}-x^{2}}}{a^{2}x}}+C}$

R = | a | x 2 + b x + c ( a ≠ 0 ) {\displaystyle R={\sqrt {|a|x^{2}+bx+c}}\;(a\neq 0)} を含む積分

${\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln \left(2{\sqrt {a}}R+2ax+b\right)\qquad ({\mbox{for }}a>0)}$

${\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\,\operatorname {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{R}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(for }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{, }}\left(2ax+b\right)<{\sqrt {b^{2}-4ac}}{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{R^{3}}}={\frac {4ax+2b}{(4ac-b^{2})R}}}$

${\displaystyle \int {\frac {dx}{R^{5}}}={\frac {4ax+2b}{3(4ac-b^{2})R}}\left({\frac {1}{R^{2}}}+{\frac {8a}{4ac-b^{2}}}\right)}$

${\displaystyle \int {\frac {dx}{R^{2n+1}}}={\frac {2}{(2n-1)(4ac-b^{2})}}\left({\frac {2ax+b}{R^{2n-1}}}+4a(n-1)\int {\frac {dx}{R^{2n-1}}}\right)}$

${\displaystyle \int {\frac {x}{R}}\;dx={\frac {R}{a}}-{\frac {b}{2a}}\int {\frac {dx}{R}}}$

${\displaystyle \int {\frac {x}{R^{3}}}\;dx=-{\frac {2bx+4c}{(4ac-b^{2})R}}}$

${\displaystyle \int {\frac {x}{R^{2n+1}}}\;dx=-{\frac {1}{(2n-1)aR^{2n-1}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{2n+1}}}}$

${\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {c}}R+bx+2c}{x}}\right)}$

${\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\operatorname {arsinh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right)}$

三角関数を含む積分

${\displaystyle \int \cos x\,dx=\sin x+C}$

${\displaystyle \int -\sin x\,dx=\cos x+C}$

${\displaystyle \int \sec ^{2}x\,dx=\tan x+C}$

${\displaystyle \int -\csc ^{2}x\,dx=\cot x+C}$

${\displaystyle \int \sec x\tan x\,dx=\sec x+C}$

${\displaystyle \int -\csc x\cot x\,dx=\csc x+C}$

${\displaystyle \int \tan x\,dx=-\ln(\cos x)+C}$

${\displaystyle \int \cot x\,dx=\ln(\sin x)+C}$

${\displaystyle \int \sec x\,dx=\ln(\sec x+\tan x)+C=\operatorname {gd} ^{-1}x+C\quad \operatorname {gd} ^{-1}x}$：グーデルマン関数の逆関数

${\displaystyle \int \csc x\,dx=-\ln(\csc x+\cot x)+C=\ln \left({\tan x-\sin x \over \sin x\tan x}\right)+C}$

${\displaystyle \int \sin ^{n}x\,dx=-{\frac {1}{n}}\sin ^{n-1}x\cos x+{\frac {n-1}{n}}\int \sin ^{n-2}x\,dx+C\quad \forall n\geq 2}$

${\displaystyle \int \sin ^{2}x\,dx={\frac {x}{2}}-{\frac {\sin {2x}}{4}}+C}$

${\displaystyle \int \cos ^{n}x\,dx={\frac {1}{n}}\cos ^{n-1}x\sin x+{\frac {n-1}{n}}\int \cos ^{n-2}x\,dx+C\quad \forall n\geq 2}$

${\displaystyle \int \cos ^{2}x\,dx={\frac {x}{2}}+{\frac {\sin {2x}}{4}}+C}$

${\displaystyle \int \tan ^{n}x\,dx={\frac {1}{n-1}}\tan ^{n-1}x-\int \tan ^{n-2}x\,dx+C\quad \forall n\geq 2}$

${\displaystyle \int \tan ^{2}x\,dx=\tan x-x+C}$

${\displaystyle \int \cot ^{n}x\,dx=-{\frac {1}{n-1}}\cot ^{n-1}x-\int \cot ^{n-2}x\,dx+C\quad \forall n\geq 2}$

${\displaystyle \int \cot ^{2}x\,dx=-\cot x-x+C}$

${\displaystyle \int \sec ^{n}x\,dx={\frac {1}{n-1}}\sec ^{n-2}x\tan x+{\frac {n-2}{n-1}}\int \sec ^{n-2}x\,dx+C\quad \forall n\geq 2}$

${\displaystyle \int \csc ^{n}x\,dx=-{\frac {1}{n-1}}\csc ^{n-2}x\cot x+{\frac {n-2}{n-1}}\int \csc ^{n-2}x\,dx+C\quad \forall n\geq 2}$

逆三角関数を含む積分

${\displaystyle \int \arcsin x\,dx=x\arcsin x+{\sqrt {1-x^{2}}}+C}$

${\displaystyle \int \arccos x\,dx=x\arccos x-{\sqrt {1-x^{2}}}+C}$

${\displaystyle \int \arctan x\,dx=x\arctan x-\ln {\sqrt {1+x^{2}}}+C}$

${\displaystyle \int \operatorname {arccot} x\,dx=x\operatorname {arccot} x+\ln {\sqrt {1+x^{2}}}+C}$

${\displaystyle \int \operatorname {arcsec} x\,dx=x\operatorname {arcsec} x-\ln(x-{\sqrt {x^{2}-1}})+C}$

${\displaystyle \int \operatorname {arccsc} x\,dx=x\operatorname {arccsc} x+\ln(x+{\sqrt {x^{2}-1}})+C}$

指数関数を含む積分

${\displaystyle \int e^{x}\,dx=e^{x}+C}$

${\displaystyle \int \alpha ^{x}\,dx={\frac {\alpha ^{x}}{\ln \alpha }}+C}$

${\displaystyle \int xe^{ax}\,dx={\frac {1}{a^{2}}}(ax-1)e^{ax}+C}$

${\displaystyle \int x^{n}e^{ax}\,dx={\frac {1}{a}}x^{n}e^{ax}-{\frac {n}{a}}\int x^{n-1}e^{ax}\,dx}$

${\displaystyle \int e^{ax}\sin bx\,dx={\frac {e^{ax}}{a^{2}+b^{2}}}(a\sin bx-b\cos bx)+C}$

${\displaystyle \int e^{ax}\cos bx\,dx={\frac {e^{ax}}{a^{2}+b^{2}}}(a\cos bx+b\sin bx)+C}$

対数関数を含む積分

${\displaystyle \int \ln x\,dx=x\ln x-x+C}$

${\displaystyle \int \log _{\alpha }x\,dx={\frac {1}{\ln \alpha }}\left({x\ln x-x}\right)+C}$

${\displaystyle \int x^{n}\ln x\,dx={\frac {x^{n+1}}{(n+1)^{2}}}[(n+1)\ln x-1]+C}$

${\displaystyle \int {\frac {1}{x\ln {x}}}\,dx=\ln {(\ln {x})}+C}$

双曲線関数を含む積分

${\displaystyle \int \sinh x\,dx=\cosh x+C}$

${\displaystyle \int \cosh x\,dx=\sinh x+C}$

${\displaystyle \int \tanh x\,dx=\ln \left(\cosh x\right)+C}$

${\displaystyle \int \coth x\,dx=\ln \left(\sinh x\right)+C}$

${\displaystyle \int {\mbox{sech}}\ x\,dx=\arcsin \left(\tanh x\right)+C=\arctan \left(\sinh x\right)+C=\operatorname {gd} x+C\quad \operatorname {gd} x}$：グーデルマン関数

${\displaystyle \int {\mbox{csch}}\ x\,dx=\ln \left(\tanh {x \over 2}\right)+C}$

定積分

${\displaystyle \int _{-\infty }^{\infty }e^{-\alpha x^{2}}\,dx={\sqrt {\frac {\pi }{\alpha }}}}$

${\displaystyle \int _{0}^{\frac {\pi }{2}}{\mbox{sin}}^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}{\mbox{cos}}^{n}x\,dx={\begin{cases}{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdot \cdots \cdot {\frac {4}{5}}\cdot {\frac {2}{3}},&{\mbox{if }}n>1{\mbox{ and }}n{\mbox{ is odd}}\\{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdot \cdots \cdot {\frac {3}{4}}\cdot {\frac {1}{2}}\cdot {\frac {\pi }{2}},&{\mbox{if }}n>0{\mbox{ and }}n{\mbox{ is even}}\end{cases}}}$

関連項目

• 微分積分学の基礎定理
• 解析学