指数関数の原始関数の一覧
∫ 0 1 e x ⋅ ln  a + ( 1 − x ) ⋅ ln  b d x = ∫ 0 1 ( a b ) x ⋅ b d x = ∫ 0 1 a x ⋅ b 1 − x d x = a − b ln  a − ln  b {\displaystyle \int _{0}^{1}e^{x\cdot \ln a+(1-x)\cdot \ln b}\;\mathrm {d} x=\int _{0}^{1}\left({\frac {a}{b}}\right)^{x}\cdot b\;\mathrm {d} x=\int _{0}^{1}a^{x}\cdot b^{1-x}\;\mathrm {d} x={\frac {a-b}{\ln a-\ln b}}} ( a > 0 , b > 0 , a ≠ b {\displaystyle a>0,\ b>0,\ a\neq b} )
∫ 0 ∞ e a x d x = 1 − a ( Re  ( a ) < 0 ) {\displaystyle \int _{0}^{\infty }e^{ax}\,\mathrm {d} x={\frac {1}{-a}}\quad (\operatorname {Re} (a)<0)}
∫ 0 ∞ e − a x 2 d x = 1 2 π a ( a > 0 ) {\displaystyle \int _{0}^{\infty }e^{-ax^{2}}\,\mathrm {d} x={\frac {1}{2}}{\sqrt {\pi \over a}}\quad (a>0)} (ガウス積分)