原始関数の一覧
∫ a + b x x d x = 2 a + b x + a ∫ 1 x a + b x d x {\displaystyle \int {\frac {\sqrt {a+bx}}{x}}\,dx=2{\sqrt {a+bx}}+a\int {\frac {1}{x{\sqrt {a+bx}}}}dx}
∫ a + b x x n d x = − 1 a ( n − 1 ) ( a + b x ) 3 2 x n − 1 − ( 2 n − 5 ) b 2 a ( n − 1 ) ∫ a + b x x n − 1 d x , n ≠ 1 {\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}
∫ e x d x = e x + C {\displaystyle \int e^{x}\,dx=e^{x}+C}
∫ α x d x = α x ln  α + C {\displaystyle \int \alpha ^{x}\,dx={\frac {\alpha ^{x}}{\ln \alpha }}+C}
∫ x e a x d x = 1 a 2 ( a x − 1 ) e a x + C {\displaystyle \int xe^{ax}\,dx={\frac {1}{a^{2}}}(ax-1)e^{ax}+C}
∫ x n e a x d x = 1 a x n e a x − n a ∫ x n − 1 e a x d x {\displaystyle \int x^{n}e^{ax}\,dx={\frac {1}{a}}x^{n}e^{ax}-{\frac {n}{a}}\int x^{n-1}e^{ax}\,dx}
∫ e a x sin  b x d x = e a x a 2 + b 2 ( a sin  b x − b cos  b x ) + C {\displaystyle \int e^{ax}\sin bx\,dx={\frac {e^{ax}}{a^{2}+b^{2}}}(a\sin bx-b\cos bx)+C}
∫ e a x cos  b x d x = e a x a 2 + b 2 ( a cos  b x + b sin  b x ) + C {\displaystyle \int e^{ax}\cos bx\,dx={\frac {e^{ax}}{a^{2}+b^{2}}}(a\cos bx+b\sin bx)+C}
∫ cos n  x d x = 1 n cos n − 1  x sin  x + n − 1 n ∫ cos n − 2  x d x + C ∀ n ≥ 2 {\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}
∫ cos 2  x d x = x 2 + sin  2 x 4 + C {\displaystyle \int \cos ^{2}x\,dx={\frac {x}{2}}+{\frac {\sin {2x}}{4}}+C}
∫ tan n  x d x = 1 n − 1 tan n − 1  x − ∫ tan n − 2  x d x + C ∀ n ≥ 2 {\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}
∫ tan 2  x d x = tan  x − x + C {\displaystyle \int \tan ^{2}x\,dx=\tan x-x+C}
∫ sinh  x d x = cosh  x + C {\displaystyle \int \sinh x\,dx=\cosh x+C}
∫ cosh  x d x = sinh  x + C {\displaystyle \int \cosh x\,dx=\sinh x+C}
∫ tanh  x d x = ln  ( cosh  x ) + C {\displaystyle \int \tanh x\,dx=\ln \left(\cosh x\right)+C}
∫ coth  x d x = ln  ( sinh  x ) + C {\displaystyle \int \coth x\,dx=\ln \left(\sinh x\right)+C}
∫ sech x d x = arcsin  ( tanh  x ) + C = arctan  ( sinh  x ) + C = gd  x + C gd  x {\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
∫ csch x d x = ln  ( tanh  x 2 ) + C {\displaystyle \int {\mbox{csch}}\ x\,dx=\ln \left(\tanh {x \over 2}\right)+C}