ガウス関数の原始関数の一覧
∫ 0 ∞ x φ ( x ) Φ ( b x ) d x = 1 2 2 π ( 1 + b 1 + b 2 ) {\displaystyle \int _{0}^{\infty }x\phi (x)\Phi (bx)\,dx={\frac {1}{2{\sqrt {2\pi }}}}\left(1+{\frac {b}{\sqrt {1+b^{2}}}}\right)}
∫ 0 ∞ x 2 φ ( x ) Φ ( b x ) d x = 1 4 + 1 2 π ( b 1 + b 2 + arctan  ( b ) ) {\displaystyle \int _{0}^{\infty }x^{2}\phi (x)\Phi (bx)\,dx={\frac {1}{4}}+{\frac {1}{2\pi }}\left({\frac {b}{1+b^{2}}}+\arctan(b)\right)}
∫ 0 ∞ x φ ( x ) 2 Φ ( x ) d x = 1 4 π 3 {\displaystyle \int _{0}^{\infty }x\phi (x)^{2}\Phi (x)\,dx={\frac {1}{4\pi {\sqrt {3}}}}}
∫ 0 ∞ Φ ( b x ) 2 φ ( x ) d x = 1 2 π ( arctan  ( b ) + arctan  1 + 2 b 2 ) {\displaystyle \int _{0}^{\infty }\Phi (bx)^{2}\phi (x)\,dx={\frac {1}{2\pi }}\left(\arctan(b)+\arctan {\sqrt {1+2b^{2}}}\right)}
n!! は二重階乗である。
∫ φ ( x ) d x = Φ ( x ) + C {\displaystyle \int \phi (x)\,dx=\Phi (x)+C}
∫ x φ ( x ) d x = − φ ( x ) + C {\displaystyle \int x\phi (x)\,dx=-\phi (x)+C}
∫ x 2 φ ( x ) d x = Φ ( x ) − x φ ( x ) + C {\displaystyle \int x^{2}\phi (x)\,dx=\Phi (x)-x\phi (x)+C}
∫ x 2 k + 1 φ ( x ) d x = − φ ( x ) ∑ j = 0 k ( 2 k ) ! ! ( 2 j ) ! ! x 2 j + C {\displaystyle \int x^{2k+1}\phi (x)\,dx=-\phi (x)\sum _{j=0}^{k}{\frac {(2k)!!}{(2j)!!}}x^{2j}+C}
∫ x 2 k + 2 φ ( x ) d x = − φ ( x ) ∑ j = 0 k ( 2 k + 1 ) ! ! ( 2 j + 1 ) ! ! x 2 j + 1 + ( 2 k + 1 ) ! ! Φ ( x ) + C {\displaystyle \int x^{2k+2}\phi (x)\,dx=-\phi (x)\sum _{j=0}^{k}{\frac {(2k+1)!!}{(2j+1)!!