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# 対数関数の原始関数の一覧

## 対数関数のみ

${\displaystyle \int \ln ax\;dx=x\ln ax-x}$
${\displaystyle \int \ln(ax+b)\;dx={\frac {(ax+b)\ln(ax+b)-(ax+b)}{a}}}$
${\displaystyle \int (\ln x)^{2}\;dx=x(\ln x)^{2}-2x\ln x+2x}$
${\displaystyle \int (\ln x)^{n}\;dx=x\sum _{k=0}^{n}(-1)^{n-k}{\frac {n!}{k!}}(\ln x)^{k}}$
${\displaystyle \int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{k=2}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}}$
${\displaystyle \int {\frac {dx}{(\ln x)^{n}}}=-{\frac {x}{(n-1)(\ln x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {dx}{(\ln x)^{n-1}}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}$
${\displaystyle \int (\ln x)^{x}\;dx=(\ln x)^{x-1}+\left(\ln(\ln x)\right)(\ln x)^{x}}$

## 変数の冪を含む積分

${\displaystyle \int x^{m}\ln x\;dx=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)\qquad {\mbox{(for }}m\neq -1{\mbox{)}}}$
${\displaystyle \int x^{m}(\ln x)^{n}\;dx={\frac {x^{m+1}(\ln x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\ln x)^{n-1}dx\qquad {\mbox{(for }}m\neq -1{\mbox{)}}}$
${\displaystyle \int {\frac {\left(\ln x\right)^{n}\;dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}$
${\displaystyle \int {\frac {\ln {x^{n}}\;dx}{x}}={\frac {\left(\ln {x^{n}}\right)^{2}}{2n}}\qquad {\mbox{(for }}n\neq 0{\mbox{)}}}$
${\displaystyle \int {\frac {\ln x\,dx}{x^{m}}}=-{\frac {\ln x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}}$
${\displaystyle \int {\frac {(\ln x)^{n}\;dx}{x^{m}}}=-{\frac {(\ln x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\ln x)^{n-1}dx}{x^{m}}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}}$
${\displaystyle \int {\frac {x^{m}\;dx}{(\ln x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}dx}{(\ln x)^{n-1}}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}$
${\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|}$
${\displaystyle \int {\frac {dx}{x^{n}\ln x}}=\ln \left|\ln x\right|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(n-1)^{k}(\ln x)^{k}}{k\cdot k!}}}$
${\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}$
${\displaystyle \int \ln \left(x^{2}+a^{2}\right)\;dx=x\ln \left(x^{2}+a^{2}\right)-2x+2a\tan ^{-1}{\frac {x}{a}}}$
${\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\ln \left(x^{2}+a^{2}\right)\;dx={\frac {1}{4}}\ln ^{2}\left(x^{2}+a^{2}\right)}$

## 三角関数を含む積分

${\displaystyle \int \sin(\ln x)\;dx={\frac {x}{2}}\left(\sin(\ln x)-\cos(\ln x)\right)}$
${\displaystyle \int \cos(\ln x)\;dx={\frac {x}{2}}\left(\sin(\ln x)+\cos(\ln x)\right)}$

## 指数関数を含む積分

${\displaystyle \int e^{x}\left(x\ln x-x-{\frac {1}{x}}\right)\;dx=e^{x}(x\ln x-x-\ln x)}$
${\displaystyle \int {\frac {1}{e^{x}}}\left({\frac {1}{x}}-\ln x\right)\;dx={\frac {\ln x}{e^{x}}}}$
${\displaystyle \int e^{x}\left({\frac {1}{\ln x}}-{\frac {1}{x\ln ^{2}x}}\right)\;dx={\frac {e^{x}}{\ln x}}}$

## 高階積分

${\displaystyle \underbrace {{\biggl .}{\biggr .}\int \cdots \int } _{n}\,\ln x\,\underbrace {{\biggl .}{\biggr .}{\,\mathrm {d} }x\cdots {\,\mathrm {d} }x} _{n}={\frac {x^{n}}{n!}}\left(\ln \,x-\sum _{k=1}^{n}{\frac {1}{k}}\right)+\sum _{k=0}^{n-1}C_{k}{\frac {x^{k}}{k!}}}$

## 出典

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover (1965) ISBN 978-0486612720　いくつかの積分がこの古典的書籍のpage 69 に掲載されている。