ガウス関数の原始関数の一覧
∫ x 2 φ ( a + b x ) d x = 1 b 3 ( ( a 2 + 1 ) Φ ( a + b x ) + ( a − b x ) φ ( a + b x ) ) + C {\displaystyle \int x^{2}\phi (a+bx)\,dx={\frac {1}{b^{3}}}\left((a^{2}+1)\Phi (a+bx)+(a-bx)\phi (a+bx)\right)+C}
∫ φ ( a + b x ) n d x = 1 b n ( 2 π ) n − 1 Φ ( n ( a + b x ) ) + C {\displaystyle \int \phi (a+bx)^{n}\,dx={\frac {1}{b{\sqrt {n(2\pi )^{n-1}}}}}\Phi \left({\sqrt {n}}(a+bx)\right)+C}
∫ Φ ( a + b x ) d x = 1 b ( ( a + b x ) Φ ( a + b x ) + φ ( a + b x ) ) + C {\displaystyle \int \Phi (a+bx)\,dx={\frac {1}{b}}\left((a+bx)\Phi (a+bx)+\phi (a+bx)\right)+C}
∫ x Φ ( a + b x ) d x = 1 2 b 2 ( ( b 2 x 2 − a 2 − 1 ) Φ ( a + b x ) + ( b x − a ) φ ( a + b x ) ) + C {\displaystyle \int x\Phi (a+bx)\,dx={\frac {1}{2b^{2}}}\left((b^{2}x^{2}-a^{2}-1)\Phi (a+bx)+(bx-a)\phi (a+bx)\right)+C}