指数関数の原始関数の一覧
∫ 0 ∞ x e − a x sin  b x d x = 2 a b ( a 2 + b 2 ) 2 ( a > 0 ) {\displaystyle \int _{0}^{\infty }xe^{-ax}\sin bx\,\mathrm {d} x={\frac {2ab}{(a^{2}+b^{2})^{2}}}\quad (a>0)}
∫ 0 ∞ x e − a x cos  b x d x = a 2 − b 2 ( a 2 + b 2 ) 2 ( a > 0 ) {\displaystyle \int _{0}^{\infty }xe^{-ax}\cos bx\,\mathrm {d} x={\frac {a^{2}-b^{2}}{(a^{2}+b^{2})^{2}}}\quad (a>0)}
∫ 0 2 π e x cos  θ d θ = 2 π I 0 ( x ) {\displaystyle \int _{0}^{2\pi }e^{x\cos \theta }d\theta =2\pi I_{0}(x)} ( I 0 {\displaystyle I_{0}} は変形ベッセル関数)
∫ 0 2 π e x cos  θ + y sin  θ d θ = 2 π I 0 ( x 2 + y 2 ) {\displaystyle \int _{0}^{2\pi }e^{x\cos \theta +y\sin \theta }d\theta =2\pi I_{0}\left({\sqrt {x^{2}+y^{2}}}\right)}
∫ e c x ln  x d x = 1 c ( e c x ln  | x | − Ei ( c x ) ) {\displaystyle \int e^{cx}\ln x\;\mathrm {d} x={\frac {1}{c}}\left(e^{cx}\ln |x|-\operatorname {Ei} \,(cx)\right)}
∫ e c x sin  b x d x = e c x c 2 + b 2 ( c sin  b x − b cos  b x ) {\displaystyle \int e^{cx}\sin bx\;\mathrm {d} x={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)}
∫ e c x cos  b x d x = e c x c 2 + b 2 ( c cos  b x + b sin  b x ) {\displaystyle \int e^{cx}\cos bx\;\mathrm {d} x={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)}