ガウス関数の原始関数の一覧
}}x^{2j+1}+(2k+1)!!\,\Phi (x)+C}
∫ φ ( x ) 2 d x = 1 2 π Φ ( x 2 ) + C {\displaystyle \int \phi (x)^{2}\,dx={\frac {1}{2{\sqrt {\pi }}}}\Phi \left(x{\sqrt {2}}\right)+C}
∫ φ ( x ) φ ( a + b x ) d x = 1 t φ ( a t ) Φ ( t x + a b t ) + C , t = 1 + b 2 {\displaystyle \int \phi (x)\phi (a+bx)\,dx={\frac {1}{t}}\phi \left({\frac {a}{t}}\right)\Phi \left(tx+{\frac {ab}{t}}\right)+C,\qquad t={\sqrt {1+b^{2}}}}
∫ x φ ( a + b x ) d x = − 1 b 2 ( φ ( a + b x ) + a Φ ( a + b x ) ) + C {\displaystyle \int x\phi (a+bx)\,dx=-{\frac {1}{b^{2}}}\left(\phi (a+bx)+a\Phi (a+bx)\right)+C}
∫ − ∞ ∞ Φ ( a + b x ) 2 φ ( x ) d x = Φ ( a 1 + b 2 ) − 2 T ( a 1 + b 2 , 1 1 + 2 b 2 ) {\displaystyle \int _{-\infty }^{\infty }\Phi (a+bx)^{2}\phi (x)\,dx=\Phi \left({\frac {a}{\sqrt {1+b^{2}}}}\right)-2T\left({\frac {a}{\sqrt {1+b^{2}}}},{\frac {1}{\sqrt {1+2b^{2}}}}\right)}
∫ − ∞ ∞ x Φ ( a + b x ) 2 φ ( x ) d x = 2 b 1 + b 2 φ ( a t ) Φ ( a 1 + b 2 1 + 2 b 2 ) {\displaystyle \int _{-\infty }^{\infty }x\Phi (a+bx)^{2}\phi (x)\,dx={\frac {2b}{\sqrt {1+b^{2}}}}\phi \left({\frac {a}{t}}\right)\Phi \left({\frac {a}{{\sqrt {1+b^{2}}}{\sqrt {1+2b^{2}}}}}\right)}