原始関数の一覧
∫ 1 a x 2 + b d x = 1 a b arctan  a b x + C {\displaystyle \int {\frac {1}{ax^{2}+b}}\,dx={\frac {1}{\sqrt {ab}}}\arctan {\sqrt {\frac {a}{b}}}x+C}
∫ a 2 − x 2 x 2 d x = − a 2 − x 2 x − arcsin  x a + C {\displaystyle \int {\frac {\sqrt {a^{2}-x^{2}}}{x^{2}}}\,dx=-{\frac {\sqrt {a^{2}-x^{2}}}{x}}-\arcsin {\frac {x}{a}}+C}
∫ 1 x a 2 − x 2 d x = − 1 a ln  ( a + a 2 − x 2 x ) + 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}
∫ x 2 a 2 − x 2 d x = − 1 2 x a 2 − x 2 + 1 2 a 2 arcsin  x a + 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}
∫ 1 x 2 a 2 − x 2 d x = − a 2 − x 2 a 2 x + C {\displaystyle \int {\frac {1}{x^{2}{\sqrt {a^{2}-x^{2}}}}}\,dx=-{\frac {\sqrt {a^{2}-x^{2}}}{a^{2}x}}+C}
∫ 1 a 2 − x 2 d x = arcsin  x a + C = − arccos  x a + C {\displaystyle \int {\frac {1}{\sqrt {a^{2}-x^{2}}}}\,dx=\arcsin {\frac {x}{a}}+C=-\arccos {\frac {x}{a}}+C}
∫ a 2 − x 2 d x = 1 2 x a 2 − x 2 + a 2 2 arcsin  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}
∫ x 2 a 2 − x 2 d x = 1 8 x ( 2 x 2 − a 2 ) a 2 − x 2 + 1 8 a 4 arcsin  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}
∫ a 2 − x 2 x d x = a 2 − x 2 − a ln  ( a + a 2 − x 2 x ) + 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}
∫ − ∞ ∞ e − α x 2 d x = π α {\displaystyle \int _{-\infty }^{\infty }e^{-\alpha x^{2}}\,dx={\sqrt {\frac {\pi }{\alpha }}}}
∫ 0 π 2 sin n x d x = ∫ 0 π 2 cos n x d x = { n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ ⋅ 4 5 ⋅ 2 3 , if n > 1 and n is odd n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ ⋅ 3 4 ⋅ 1 2 ⋅ π 2 , if n > 0 and n is even {\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}}}