指数関数の原始関数の一覧

本稿は指数関数を含む式の原始関数の一覧である。 以下は不定積分の一覧である。右辺につく積分定数は省略している。 ∫ e x d x = e x {\displaystyle \int e^{x}\;\mathrm {d} x=e^{x}} ∫ f ′ ( x ) e f ( x ) d x = e f ( x ) {\displaystyle \int f'(x)e^{f(x)}\;\mathrm {d} x=e^{f(x)}} ∫ e c x d x = 1 c e c x {\displaystyle \int e^{cx}\;\mathrm {d} x={\frac {1}{c}}e^{cx}} ∫ a c x d x = 1 c ⋅ ln ⁡ a a c x {\displaystyle \int a^{cx}\;\mathrm {d} x={\frac {1}{c\cdot \ln a}}a^{cx}} （ a > 0 ,   a ≠ 1 {\displaystyle a>0,\ a\neq 1} ） ∫ x e c x d x = e c x c 2 ( c x − 1 ) {\displaystyle \int xe^{cx}\;\mathrm {d} x={\frac {e^{cx}}{c^{2}}}(cx-1)} ∫ x 2 e c x d x = e c x ( x 2 c − 2 x c 2 + 2 c 3 ) {\displaystyle \int x^{2}e^{cx}\;\mathrm {d} x=e^{cx}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)} ∫ x n e c x d x = 1 c x n e c x − n c ∫ x n − 1 e c x d x = ( ∂ ∂ c ) n e c x c {\displaystyle \int x^{n}e^{cx}\;\mathrm {d} x={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}\mathrm {d} x=\left({\frac {\partial }{\partial c}}\right)^{n}{\frac {e^{cx}}{c}}} ∫ e c x x d x = ln ⁡ | x | + ∑ n = 1 ∞ ( c x ) n n ⋅ n ! {\displaystyle \int {\frac {e^{cx}}{x}}\;\mathrm {d} x=\ln |x|+\sum _{n=1}^{\infty }{\frac {(cx)^{n}}{n\cdot n!}}} ∫ e c x x n d x = 1 n − 1 ( − e c x x n − 1 + c ∫ e c x x n − 1 d x ) ( n ≠ 1 ) {\displaystyle \int {\frac {e^{cx}}{x^{n}}}\;\mathrm {d} x={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}}{x^{n-1}}}\,\mathrm {d} x\right)\qquad {\mbox{(}}n\neq 1{\mbox{)}}} ∫ 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)} ∫ e c x sin n ⁡ x d x = e c x sin n − 1 ⁡ x c 2 + n 2 ( c sin ⁡ x − n cos ⁡ x ) + n ( n − 1 ) c 2 + n 2 ∫ e c x sin n − 2 ⁡ x d x {\displaystyle \int e^{cx}\sin ^{n}x\;\mathrm {d} x={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\sin ^{n-2}x\;\mathrm {d} x} ∫ e c x cos n ⁡ x d x = e c x cos n − 1 ⁡ x c 2 + n 2 ( c cos ⁡ x + n sin ⁡ x ) + n ( n − 1 ) c 2 + n 2 ∫ e c x cos n − 2 ⁡ x d x {\displaystyle \int e^{cx}\cos ^{n}x\;\mathrm {d} x={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\cos ^{n-2}x\;\mathrm {d} x} ∫ x e c x 2 d x = 1 2 c e c x 2 {\displaystyle \int xe^{cx^{2}}\;\mathrm {d} x={\frac {1}{2c}}\;e^{cx^{2}}} ∫ e − c x 2 d x = π 4 c erf ⁡ ( c x ) {\displaystyle \int e^{-cx^{2}}\;\mathrm {d} x={\sqrt {\frac {\pi }{4c}}}\operatorname {erf} ({\sqrt {c}}x)} （ erf {\displaystyle \operatorname {erf} } は誤差関数） ∫ x e − c x 2 d x = − 1 2 c e − c x 2 {\displaystyle \int xe^{-cx^{2}}\;\mathrm {d} x=-{\frac {1}{2c}}e^{-cx^{2}}} ∫ e − x 2 x 2 d x = − e − x 2 x − π e r f ( x ) {\displaystyle \int {\frac {e^{-x^{2}}}{x^{2}}}\;\mathrm {d} x=-{\frac {e^{-x^{2}}}{x}}-{\sqrt {\pi }}\mathrm {erf} (x)} ∫ 1 σ 2 π e − 1 2 ( x − μ σ ) 2 d x = 1 2 ( erf x − μ σ 2 ) {\displaystyle \int {{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}\;\mathrm {d} x={\frac {1}{2}}\left(\operatorname {erf} \,{\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)} ∫ e x 2 d x = e x 2 ( ∑ j = 0 n − 1 c 2 j 1 x 2 j + 1 ) + ( 2 n − 1 ) c 2 n − 2 ∫ e x 2 x 2 n d x ( n > 0 ) {\displaystyle \int e^{x^{2}}\,\mathrm {d} x=e^{x^{2}}\left(\sum _{j=0}^{n-1}c_{2j}\,{\frac {1}{x^{2j+1}}}\right)+(2n-1)c_{2n-2}\int {\frac {e^{x^{2}}}{x^{2n}}}\;\mathrm {d} x\quad {\mbox{(}}n>0{\mbox{)}}} ここで c 2 j = 1 ⋅ 3 ⋅ 5 ⋯ ( 2 j − 1 ) 2 j + 1 = ( 2 j ) ! j ! 2 2 j + 1 {\displaystyle c_{2j}={\frac {1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}}}={\frac {(2j)\,!}{j!\,2^{2j+1}}}} とする。 ∫ x x ⋅ ⋅ x ⏟ m d x = ∫ x ↑↑ m d x = ∫ x → m → 2 d x = ∑ n = 0 m ( − 1 ) n ( n + 1 ) n − 1 n ! 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 ∫ 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\,.} ） ∫ 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） ∫ − ∞ ∞ 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)} ∫ 0 ∞ x n e − a x 2 d x = { 1 2 Γ ( n + 1 2 ) / a n + 1 2 ( n > − 1 , a > 0 ) ( 2 k − 1 ) ! ! 2 k + 1 a k π a ( n = 2 k , k integer , a > 0 ) k ! 2 a k + 1 ( n = 2 k + 1 , k integer , a > 0 ) {\displaystyle \int _{0}^{\infty }x^{n}e^{-ax^{2}}\,\mathrm {d} x={\begin{cases}{\frac {1}{2}}\Gamma \left({\frac {n+1}{2}}\right)/a^{\frac {n+1}{2}}&(n>-1,a>0)\\{\frac {(2k-1)!!}{2^{k+1}a^{k}}}{\sqrt {\frac {\pi }{a}}}&(n=2k,k\;{\text{integer}},a>0)\\{\frac {k!}{2a^{k+1}}}&(n=2k+1,k\;{\text{integer}},a>0)\end{cases}}} （!! は二重階乗） ∫ 0 ∞ x n e − a x d x = { Γ ( n + 1 ) a n + 1 ( n > − 1 , a > 0 ) n ! a n + 1 ( n = 0 , 1 , 2 , … , a > 0 ) {\displaystyle \int _{0}^{\infty }x^{n}e^{-ax}\,\mathrm {d} x={\begin{cases}{\frac {\Gamma (n+1)}{a^{n+1}}}&(n>-1,a>0)\\{\frac {n!}{a^{n+1}}}&(n=0,1,2,\ldots ,a>0)\\\end{cases}}} ∫ 0 ∞ e − a x b d x = 1 b   a − 1 b Γ ( 1 b ) {\displaystyle \int _{0}^{\infty }e^{-ax^{b}}dx={\frac {1}{b}}\ a^{-{\frac {1}{b}}}\,\Gamma \left({\frac {1}{b}}\right)} ∫ 0 ∞ x n e − a x b d x = 1 b   a − n + 1 b Γ ( n + 1 b ) {\displaystyle \int _{0}^{\infty }x^{n}e^{-ax^{b}}dx={\frac {1}{b}}\ a^{-{\frac {n+1}{b}}}\,\Gamma \left({\frac {n+1}{b}}\right)} ∫ 0 ∞ e − a x sin ⁡ b x d x = b a 2 + b 2 ( a > 0 ) {\displaystyle \int _{0}^{\infty }e^{-ax}\sin bx\,\mathrm {d} x={\frac {b}{a^{2}+b^{2}}}\quad (a>0)} ∫ 0 ∞ e − a x cos ⁡ b x d x = a a 2 + b 2 ( a > 0 ) {\displaystyle \int _{0}^{\infty }e^{-ax}\cos bx\,\mathrm {d} x={\frac {a}{a^{2}+b^{2}}}\quad (a>0)} ∫ 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)} Wolfram Mathematica Online Integrator V. H. Moll, The Integrals in Gradshteyn and Ryzhik