ガウス関数の原始関数の一覧
∫ 0 ∞ x Φ ( a + b x ) φ ( x ) d x = b t φ ( a t ) Φ ( − a b t ) + 1 2 π Φ ( a ) , t = 1 + b 2 {\displaystyle \int _{0}^{\infty }x\Phi (a+bx)\phi (x)\,dx={\frac {b}{t}}\phi \left({\frac {a}{t}}\right)\Phi \left(-{\frac {ab}{t}}\right)+{\frac {1}{\sqrt {2\pi }}}\Phi (a),\qquad t={\sqrt {1+b^{2}}}}
∫ − ∞ ∞ ln  ( x 2 ) 1 σ φ ( x σ ) d x = ln  ( σ 2 ) − γ − ln  2 ≈ ln  ( σ 2 ) − 1.27036 {\displaystyle \int _{-\infty }^{\infty }\ln(x^{2}){\frac {1}{\sigma }}\phi \left({\frac {x}{\sigma }}\right)\,dx=\ln(\sigma ^{2})-\gamma -\ln 2\approx \ln(\sigma ^{2})-1.27036}