指数関数の原始関数の一覧

出典: フリー百科事典『ウィキペディア(Wikipedia)』

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。

${\displaystyle \int e^{x}\;\mathrm {d} x=e^{x}}${\displaystyle \int e^{x}\;\mathrm {d} x=e^{x}}

${\displaystyle \int f'(x)e^{f(x)}\;\mathrm {d} x=e^{f(x)}}${\displaystyle \int f'(x)e^{f(x)}\;\mathrm {d} x=e^{f(x)}}

${\displaystyle \int e^{cx}\;\mathrm {d} x={\frac {1}{c}}e^{cx}}${\displaystyle \int e^{cx}\;\mathrm {d} x={\frac {1}{c}}e^{cx}}

${\displaystyle \int a^{cx}\;\mathrm {d} x={\frac {1}{c\cdot \ln a}}a^{cx}}${\displaystyle \int a^{cx}\;\mathrm {d} x={\frac {1}{c\cdot \ln a}}a^{cx}} (${\displaystyle a>0,\ a\neq 1}${\displaystyle a>0,\ a\neq 1})

${\displaystyle \int xe^{cx}\;\mathrm {d} x={\frac {e^{cx}}{c^{2}}}(cx-1)}${\displaystyle \int xe^{cx}\;\mathrm {d} x={\frac {e^{cx}}{c^{2}}}(cx-1)}

${\displaystyle \int x^{2}e^{cx}\;\mathrm {d} x=e^{cx}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)}${\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)}

${\displaystyle \int x^{n}e^{cx}\;\mathrm {d} x={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}\mathrm {d} x=\left({\frac {\partial }{\partial c}}\right)^{n}{\frac {e^{cx}}{c}}}${\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}}}

${\displaystyle \int {\frac {e^{cx}}{x}}\;\mathrm {d} x=\ln |x|+\sum _{n=1}^{\infty }{\frac {(cx)^{n}}{n\cdot n!}}}${\displaystyle \int {\frac {e^{cx}}{x}}\;\mathrm {d} x=\ln |x|+\sum _{n=1}^{\infty }{\frac {(cx)^{n}}{n\cdot n!}}}

${\displaystyle \int {\frac {e^{cx}}{x^{n}}}\;\mathrm {d} x={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}}{x^{n-1}}},円\mathrm {d} x\right)\qquad {\mbox{(}}n\neq 1{\mbox{)}}}${\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}}},円\mathrm {d} x\right)\qquad {\mbox{(}}n\neq 1{\mbox{)}}}

${\displaystyle \int e^{cx}\ln x\;\mathrm {d} x={\frac {1}{c}}\left(e^{cx}\ln |x|-\operatorname {Ei} ,円(cx)\right)}${\displaystyle \int e^{cx}\ln x\;\mathrm {d} x={\frac {1}{c}}\left(e^{cx}\ln |x|-\operatorname {Ei} ,円(cx)\right)}

${\displaystyle \int e^{cx}\sin bx\;\mathrm {d} x={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)}${\displaystyle \int e^{cx}\sin bx\;\mathrm {d} x={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)}

${\displaystyle \int e^{cx}\cos bx\;\mathrm {d} x={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)}${\displaystyle \int e^{cx}\cos bx\;\mathrm {d} x={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)}

${\displaystyle \int e^{cx}\sin ^{n}x\;\mathrm {d} x={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\sin ^{n-2}x\;\mathrm {d} x}${\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}

${\displaystyle \int e^{cx}\cos ^{n}x\;\mathrm {d} x={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\cos ^{n-2}x\;\mathrm {d} x}${\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}

${\displaystyle \int xe^{cx^{2}}\;\mathrm {d} x={\frac {1}{2c}}\;e^{cx^{2}}}${\displaystyle \int xe^{cx^{2}}\;\mathrm {d} x={\frac {1}{2c}}\;e^{cx^{2}}}

${\displaystyle \int e^{-cx^{2}}\;\mathrm {d} x={\sqrt {\frac {\pi }{4c}}}\operatorname {erf} ({\sqrt {c}}x)}${\displaystyle \int e^{-cx^{2}}\;\mathrm {d} x={\sqrt {\frac {\pi }{4c}}}\operatorname {erf} ({\sqrt {c}}x)} (${\displaystyle \operatorname {erf} }${\displaystyle \operatorname {erf} } は誤差関数)

${\displaystyle \int xe^{-cx^{2}}\;\mathrm {d} x=-{\frac {1}{2c}}e^{-cx^{2}}}${\displaystyle \int xe^{-cx^{2}}\;\mathrm {d} x=-{\frac {1}{2c}}e^{-cx^{2}}}

${\displaystyle \int {\frac {e^{-x^{2}}}{x^{2}}}\;\mathrm {d} x=-{\frac {e^{-x^{2}}}{x}}-{\sqrt {\pi }}\mathrm {erf} (x)}${\displaystyle \int {\frac {e^{-x^{2}}}{x^{2}}}\;\mathrm {d} x=-{\frac {e^{-x^{2}}}{x}}-{\sqrt {\pi }}\mathrm {erf} (x)}

${\displaystyle \int {{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}\;\mathrm {d} x={\frac {1}{2}}\left(\operatorname {erf} ,円{\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)}${\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} ,円{\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)}

${\displaystyle \int e^{x^{2}},円\mathrm {d} x=e^{x^{2}}\left(\sum _{j=0}^{n-1}c_{2j},円{\frac {1}{x^{2j+1}}}\right)+(2n-1)c_{2n-2}\int {\frac {e^{x^{2}}}{x^{2n}}}\;\mathrm {d} x\quad {\mbox{(}}n>0{\mbox{)}}}${\displaystyle \int e^{x^{2}},円\mathrm {d} x=e^{x^{2}}\left(\sum _{j=0}^{n-1}c_{2j},円{\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 ${\displaystyle c_{2j}={\frac {1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}}}={\frac {(2j),円!}{j!,2円^{2j+1}}}}${\displaystyle c_{2j}={\frac {1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}}}={\frac {(2j),円!}{j!,2円^{2j+1}}}} とする。

${\displaystyle {\int \underbrace {x^{x^{\cdot ^{\cdot ^{x}}}}} _{m},円dx=\int x\uparrow \uparrow m,円dx=\int x\to m\to 2,円dx=\sum _{n=0}^{m}{\frac {(-1)^{n}(n+1)^{n-1}}{n!}}\Gamma (n+1,-\ln x)+\sum _{n=m+1}^{\infty }(-1)^{n}a_{mn}\Gamma (n+1,-\ln x)\qquad {\mbox{(for }}x>0{\mbox{)}}}}${\displaystyle {\int \underbrace {x^{x^{\cdot ^{\cdot ^{x}}}}} _{m},円dx=\int x\uparrow \uparrow m,円dx=\int x\to m\to 2,円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 ${\displaystyle a_{mn}={\begin{cases}1&n=0,\\{\frac {1}{n!}}m=1,\\{\frac {1}{n}}\sum _{j=1}^{n}ja_{m,n-j}a_{m-1,j-1}&{\text{otherwise}}\end{cases}}}${\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}}}

${\displaystyle \Gamma (x,y)}${\displaystyle \Gamma (x,y)} はガンマ関数、${\displaystyle \uparrow }${\displaystyle \uparrow } はクヌースの矢印表記、${\displaystyle \to }${\displaystyle \to } はコンウェイのチェーン表記

${\displaystyle \int {\frac {1}{ae^{\lambda x}+b}}\;\mathrm {d} x={\frac {x}{b}}-{\frac {1}{b\lambda }}\ln \left(ae^{\lambda x}+b\right),円}${\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),円} (${\displaystyle b\neq 0}${\displaystyle b\neq 0}, ${\displaystyle \lambda \neq 0}${\displaystyle \lambda \neq 0}, かつ ${\displaystyle ae^{\lambda x}+b>0,円}${\displaystyle ae^{\lambda x}+b>0,円})

${\displaystyle \int {\frac {e^{2\lambda x}}{ae^{\lambda x}+b}}\;\mathrm {d} x={\frac {1}{a^{2}\lambda }}\left[ae^{\lambda x}+b-b\ln \left(ae^{\lambda x}+b\right)\right],円}${\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],円} (${\displaystyle a\neq 0}${\displaystyle a\neq 0}, ${\displaystyle \lambda \neq 0}${\displaystyle \lambda \neq 0}, かつ ${\displaystyle ae^{\lambda x}+b>0,円.}${\displaystyle ae^{\lambda x}+b>0,円.})

${\displaystyle \int _{0}^{1}e^{x\cdot \ln a+(1-x)\cdot \ln b}\;\mathrm {d} x=\int _{0}^{1}\left({\frac {a}{b}}\right)^{x}\cdot b\;\mathrm {d} x=\int _{0}^{1}a^{x}\cdot b^{1-x}\;\mathrm {d} x={\frac {a-b}{\ln a-\ln b}}}${\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}}} (${\displaystyle a>0,\ b>0,\ a\neq b}${\displaystyle a>0,\ b>0,\ a\neq b})

${\displaystyle \int _{0}^{\infty }e^{ax},円\mathrm {d} x={\frac {1}{-a}}\quad (\operatorname {Re} (a)<0)}${\displaystyle \int _{0}^{\infty }e^{ax},円\mathrm {d} x={\frac {1}{-a}}\quad (\operatorname {Re} (a)<0)}

${\displaystyle \int _{0}^{\infty }e^{-ax^{2}},円\mathrm {d} x={\frac {1}{2}}{\sqrt {\pi \over a}}\quad (a>0)}${\displaystyle \int _{0}^{\infty }e^{-ax^{2}},円\mathrm {d} x={\frac {1}{2}}{\sqrt {\pi \over a}}\quad (a>0)} (ガウス積分)

${\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}},円\mathrm {d} x={\sqrt {\pi \over a}}\quad (a>0)}${\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}},円\mathrm {d} x={\sqrt {\pi \over a}}\quad (a>0)}

${\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}e^{-2bx},円\mathrm {d} x={\sqrt {\frac {\pi }{a}}}e^{\frac {b^{2}}{a}}\quad (a>0)}${\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}e^{-2bx},円\mathrm {d} x={\sqrt {\frac {\pi }{a}}}e^{\frac {b^{2}}{a}}\quad (a>0)} (ガウス関数の積分)

${\displaystyle \int _{-\infty }^{\infty }xe^{-a(x-b)^{2}},円\mathrm {d} x=b{\sqrt {\frac {\pi }{a}}}\quad (\operatorname {Re} (a)>0)}${\displaystyle \int _{-\infty }^{\infty }xe^{-a(x-b)^{2}},円\mathrm {d} x=b{\sqrt {\frac {\pi }{a}}}\quad (\operatorname {Re} (a)>0)}

${\displaystyle \int _{-\infty }^{\infty }x^{2}e^{-ax^{2}},円\mathrm {d} x={\frac {1}{2}}{\sqrt {\pi \over a^{3}}}\quad (a>0)}${\displaystyle \int _{-\infty }^{\infty }x^{2}e^{-ax^{2}},円\mathrm {d} x={\frac {1}{2}}{\sqrt {\pi \over a^{3}}}\quad (a>0)}

${\displaystyle \int _{0}^{\infty }x^{n}e^{-ax^{2}},円\mathrm {d} x={\begin{cases}{\frac {1}{2}}\Gamma \left({\frac {n+1}{2}}\right)/a^{\frac {n+1}{2}}&(n>-1,a>0)\\{\frac {(2k-1)!!}{2^{k+1}a^{k}}}{\sqrt {\frac {\pi }{a}}}&(n=2k,k\;{\text{integer}},a>0)\\{\frac {k!}{2a^{k+1}}}&(n=2k+1,k\;{\text{integer}},a>0)\end{cases}}}${\displaystyle \int _{0}^{\infty }x^{n}e^{-ax^{2}},円\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)

${\displaystyle \int _{0}^{\infty }x^{n}e^{-ax},円\mathrm {d} x={\begin{cases}{\frac {\Gamma (n+1)}{a^{n+1}}}&(n>-1,a>0)\\{\frac {n!}{a^{n+1}}}&(n=0,1,2,\ldots ,a>0)\\\end{cases}}}${\displaystyle \int _{0}^{\infty }x^{n}e^{-ax},円\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}}}

${\displaystyle \int _{0}^{\infty }e^{-ax^{b}}dx={\frac {1}{b}}\ a^{-{\frac {1}{b}}},円\Gamma \left({\frac {1}{b}}\right)}${\displaystyle \int _{0}^{\infty }e^{-ax^{b}}dx={\frac {1}{b}}\ a^{-{\frac {1}{b}}},円\Gamma \left({\frac {1}{b}}\right)}

${\displaystyle \int _{0}^{\infty }x^{n}e^{-ax^{b}}dx={\frac {1}{b}}\ a^{-{\frac {n+1}{b}}},円\Gamma \left({\frac {n+1}{b}}\right)}${\displaystyle \int _{0}^{\infty }x^{n}e^{-ax^{b}}dx={\frac {1}{b}}\ a^{-{\frac {n+1}{b}}},円\Gamma \left({\frac {n+1}{b}}\right)}

${\displaystyle \int _{0}^{\infty }e^{-ax}\sin bx,円\mathrm {d} x={\frac {b}{a^{2}+b^{2}}}\quad (a>0)}${\displaystyle \int _{0}^{\infty }e^{-ax}\sin bx,円\mathrm {d} x={\frac {b}{a^{2}+b^{2}}}\quad (a>0)}

${\displaystyle \int _{0}^{\infty }e^{-ax}\cos bx,円\mathrm {d} x={\frac {a}{a^{2}+b^{2}}}\quad (a>0)}${\displaystyle \int _{0}^{\infty }e^{-ax}\cos bx,円\mathrm {d} x={\frac {a}{a^{2}+b^{2}}}\quad (a>0)}

${\displaystyle \int _{0}^{\infty }xe^{-ax}\sin bx,円\mathrm {d} x={\frac {2ab}{(a^{2}+b^{2})^{2}}}\quad (a>0)}${\displaystyle \int _{0}^{\infty }xe^{-ax}\sin bx,円\mathrm {d} x={\frac {2ab}{(a^{2}+b^{2})^{2}}}\quad (a>0)}

${\displaystyle \int _{0}^{\infty }xe^{-ax}\cos bx,円\mathrm {d} x={\frac {a^{2}-b^{2}}{(a^{2}+b^{2})^{2}}}\quad (a>0)}${\displaystyle \int _{0}^{\infty }xe^{-ax}\cos bx,円\mathrm {d} x={\frac {a^{2}-b^{2}}{(a^{2}+b^{2})^{2}}}\quad (a>0)}

${\displaystyle \int _{0}^{2\pi }e^{x\cos \theta }d\theta =2\pi I_{0}(x)}${\displaystyle \int _{0}^{2\pi }e^{x\cos \theta }d\theta =2\pi I_{0}(x)} (${\displaystyle I_{0}}${\displaystyle I_{0}} は変形ベッセル関数)

${\displaystyle \int _{0}^{2\pi }e^{x\cos \theta +y\sin \theta }d\theta =2\pi I_{0}\left({\sqrt {x^{2}+y^{2}}}\right)}${\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

「https://ja.wikipedia.org/w/index.php?title=指数関数の原始関数の一覧&oldid=102966502」から取得

カテゴリ:

• 数学に関する記事
• 指数関数
• 積分法
• 数学の一覧

隠しカテゴリ:

• 出典を必要とする記事/2013年10月