ガウス関数の原始関数の一覧
∫ x φ ( x ) Φ ( a + b x ) d x = b t φ ( a t ) Φ ( x t + a b t ) − φ ( x ) Φ ( a + b x ) + C , t = 1 + b 2 {\displaystyle \int x\phi (x)\Phi (a+bx)\,dx={\frac {b}{t}}\phi \left({\frac {a}{t}}\right)\Phi \left(xt+{\frac {ab}{t}}\right)-\phi (x)\Phi (a+bx)+C,\qquad t={\sqrt {1+b^{2}}}}
∫ Φ ( x ) 2 d x = x Φ ( x ) 2 + 2 Φ ( x ) φ ( x ) − 1 π Φ ( x 2 ) + C {\displaystyle \int \Phi (x)^{2}\,dx=x\Phi (x)^{2}+2\Phi (x)\phi (x)-{\frac {1}{\sqrt {\pi }}}\Phi \left(x{\sqrt {2}}\right)+C}
∫ e c x φ ( b x ) n d x = e c 2 2 n b 2 b n ( 2 π ) n − 1 Φ ( b 2 x n − c b n ) + C , b ≠ 0 , n > 0 {\displaystyle \int e^{cx}\phi (bx)^{n}\,dx={\frac {e^{\frac {c^{2}}{2nb^{2}}}}{b{\sqrt {n(2\pi )^{n-1}}}}}\Phi \left({\frac {b^{2}xn-c}{b{\sqrt {n}}}}\right)+C,\qquad b\neq 0,n>0}