指数関数の原始関数の一覧
∫ − ∞ ∞ e − a x 2 d x = π a ( a > 0 ) {\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}\,\mathrm {d} x={\sqrt {\pi \over a}}\quad (a>0)}
∫ − ∞ ∞ e − a x 2 e − 2 b x d x = π a e b 2 a ( a > 0 ) {\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}e^{-2bx}\,\mathrm {d} x={\sqrt {\frac {\pi }{a}}}e^{\frac {b^{2}}{a}}\quad (a>0)} (ガウス関数の積分)
∫ − ∞ ∞ x e − a ( x − b ) 2 d x = b π a ( Re  ( a ) > 0 ) {\displaystyle \int _{-\infty }^{\infty }xe^{-a(x-b)^{2}}\,\mathrm {d} x=b{\sqrt {\frac {\pi }{a}}}\quad (\operatorname {Re} (a)>0)}
∫ − ∞ ∞ x 2 e − a x 2 d x = 1 2 π a 3 ( a > 0 ) {\displaystyle \int _{-\infty }^{\infty }x^{2}e^{-ax^{2}}\,\mathrm {d} x={\frac {1}{2}}{\sqrt {\pi \over a^{3}}}\quad (a>0)}
∫ 1 a e λ x + b d x = x b − 1 b λ ln  ( a e λ x + b ) {\displaystyle \int {\frac {1}{ae^{\lambda x}+b}}\;\mathrm {d} x={\frac {x}{b}}-{\frac {1}{b\lambda }}\ln \left(ae^{\lambda x}+b\right)\,} ( b ≠ 0 {\displaystyle b\neq 0} , λ ≠ 0 {\displaystyle \lambda \neq 0} , かつ a e λ x + b > 0 {\displaystyle ae^{\lambda x}+b>0\,} )
∫ e 2 λ x a e λ x + b d x = 1 a 2 λ [ a e λ x + b − b ln  ( a e λ x + b ) ] {\displaystyle \int {\frac {e^{2\lambda x}}{ae^{\lambda x}+b}}\;\mathrm {d} x={\frac {1}{a^{2}\lambda }}\left[ae^{\lambda x}+b-b\ln \left(ae^{\lambda x}+b\right)\right]\,} ( a ≠ 0 {\displaystyle a\neq 0} , λ ≠ 0 {\displaystyle \lambda \neq 0} , かつ a e λ x + b > 0 . {\displaystyle ae^{\lambda x}+b>0\,.} )
Γ ( n + 1 , − ln  x ) + ∑ n = m + 1 ∞ ( − 1 ) n a m n Γ ( n + 1 , − ln  x ) (for x > 0 ) {\displaystyle {\int \underbrace {x^{x^{\cdot ^{\cdot ^{x}}}}} _{m}\,dx=\int x\uparrow \uparrow m\,dx=\int x\to m\to 2\,dx=\sum _{n=0}^{m}{\frac {(-1)^{n}(n+1)^{n-1}}{n!}}\Gamma (n+1,-\ln x)+\sum _{n=m+1}^{\infty }(-1)^{n}a_{mn}\Gamma (n+1,-\ln x)\qquad {\mbox{(for }}x>0{\mbox{)}}}} ここで a m n = { 1 n = 0 , 1 n ! m = 1 , 1 n ∑ j = 1 n j a m , n − j a m − 1 , j − 1 otherwise {\displaystyle a_{mn}={\begin{cases}1&n=0,\\{\frac {1}{n!}}m=1,\\{\frac {1}{n}}\sum _{j=1}^{n}ja_{m,n-j}a_{m-1,j-1}&{\text{otherwise}}\end{cases}}} Γ ( x , y ) {\displaystyle \Gamma (x,y)} はガンマ関数、 ↑ {\displaystyle \uparrow } はクヌースの矢印表記、 → {\displaystyle \to } はコンウェイのチェーン表記