原始関数の一覧
∫ d x R = 1 a ln  ( 2 a R + 2 a x + b ) ( for a > 0 ) {\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln \left(2{\sqrt {a}}R+2ax+b\right)\qquad ({\mbox{for }}a>0)}
∫ d x R = 1 a arsinh  2 a x + b 4 a c − b 2 (for a > 0 , 4 a c − b 2 > 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{)}}}
∫ d x R = 1 a ln  | 2 a x + b | (for a > 0 , 4 a c − b 2 = 0 ) {\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}
∫ sec  x d x = ln  ( sec  x + tan  x ) + C = gd − 1  x + C gd − 1  x {\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
∫ csc  x d x = − ln  ( csc  x + cot  x ) + C = ln  ( tan  x − sin  x sin  x tan  x ) + C {\displaystyle \int \csc x\,dx=-\ln(\csc x+\cot x)+C=\ln \left({\tan x-\sin x \over \sin x\tan x}\right)+C}
∫ sin n  x d x = − 1 n sin n − 1  x cos  x + n − 1 n ∫ sin n − 2  x d x + C ∀ n ≥ 2 {\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}
∫ sin 2  x d x = x 2 − sin  2 x 4 + C {\displaystyle \int \sin ^{2}x\,dx={\frac {x}{2}}-{\frac {\sin {2x}}{4}}+C}
∫ 1 x 2 − a 2 d x = ln  ( x + x 2 − a 2 ) + C {\displaystyle \int {\frac {1}{\sqrt {x^{2}-a^{2}}}}\,dx=\ln \left(x+{\sqrt {x^{2}-a^{2}}}\right)+C}
∫ arcsin  x d x = x arcsin  x + 1 − x 2 + C {\displaystyle \int \arcsin x\,dx=x\arcsin x+{\sqrt {1-x^{2}}}+C}
∫ arccos  x d x = x arccos  x − 1 − x 2 + C {\displaystyle \int \arccos x\,dx=x\arccos x-{\sqrt {1-x^{2}}}+C}
∫ arctan  x d x = x arctan  x − ln  1 + x 2 + C {\displaystyle \int \arctan x\,dx=x\arctan x-\ln {\sqrt {1+x^{2}}}+C}