对数函数积分表

以下是部份对数函数的积分表(书写时省略了不定积分结果中都含有的任意常数C_n)。

注：我们规定 ${\begin{cases}\log _{\alpha }^{2}x=\log _{\alpha }\log _{\alpha }x\\\log _{\alpha }^{2}|x|=\log _{\alpha }|(\log _{\alpha }|x|)|\end{cases}}$

${\begin{cases}\int \ln \!x\ {\mbox{d}}x&=x\ln \!x-\int x\!{\dfrac {1}{x}}\ {\mbox{d}}x\\&=x\ln \!x-\int \!\!{\mbox{d}}x\\&=x\ln \!x-x\\\int \ln(-x)\ {\mbox{d}}x&=-\int \ln(-x)\ {\mbox{d}}\!(-x)\\&=-\left[\left(-x\right)\ln \left(-x\right)-\left(-x\right)\right]\\&=x\ln \left(-x\right)-x\end{cases}}\Longrightarrow {\begin{cases}\int \ln \!|x|&=x\ln \!|x|-x\\\int \log _{\alpha }\!|x|{\mbox{d}}x&=\int {\dfrac {\ln \!|x|}{\ln \!\alpha }}{\mbox{d}}x\\&={\dfrac {x\ln \!|x|-x}{\ln \!\alpha }}\\&=x\log _{\alpha }\!|x|-{\dfrac {x}{\ln \!\alpha }}\end{cases}}$

$\int \ln \!Mx{\mbox{d}}x=x\ln \!Mx-x$

$\int (\ln \!x)^{2}{\mbox{d}}x=x(\ln \!x)^{2}-2x\ln \!x+2x$

$\int (\ln \!Mx)^{n}{\mbox{d}}x=x(\ln \!Mx)^{n}-n\int (\ln \!Mx)^{n-1}dx$

$\int {\frac {{\mbox{d}}x}{\ln \!x}}=\ln \!|\!\ln \!x|+\ln \!x+\sum _{I=2}^{\infty }{\frac {(\ln \!x)^{I}}{I!I}}$

$\int {\frac {{\mbox{d}}x}{(\ln \!x)^{n}}}=-{\frac {x}{(n-1)(\ln \!x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {{\mbox{d}}x}{(\ln x)^{n-1}}}\qquad (n\neq 1)$

$\int x^{N}\ln \!x{\mbox{d}}x=x^{N+1}\left({\frac {\ln \!x}{N+1}}-{\frac {1}{(N+1)^{2}}}\right)\qquad (N\neq -1)$

$\int x^{N}(\ln \!x)^{n}\ {\mbox{d}}x={\frac {x^{N+1}(\ln \!x)^{n}}{N+1}}-{\frac {n}{N+1}}\int x^{N}(\ln x)^{n-1}dx\qquad (N\neq -1)$

$\int {\frac {(\ln \!x)^{n}\ {\mbox{d}}x}{x}}={\frac {(\ln \!x)^{n+1}}{n+1}}\qquad (n\neq -1)$

$\int {\frac {\ln \!x\ {\mbox{d}}x}{x^{r}}}=-{\frac {\ln \!x}{(r-1)x^{r-1}}}-{\frac {1}{(1-r)^{2}x^{r-1}}}\qquad (r\neq 1)$

$\int {\frac {(\ln \!x)^{n}\ {\mbox{d}}x}{x^{r}}}=-{\frac {(\ln x)^{n}}{(r-1)x^{r-1}}}+{\frac {n}{r-1}}\int {\frac {(\ln x)^{n-1}dx}{x^{m}}}\qquad (m\neq 1)$

$\int {\frac {x^{N}\ {\mbox{d}}x}{(\ln \!x)^{n}}}=-{\frac {x^{N+1}}{(n-1)(\ln \!x)^{n-1}}}+{\frac {N+1}{n-1}}\int {\frac {x^{N}{\mbox{d}}x}{(\ln \!x)^{n-1}}}\qquad (n\neq 1)$

$\int {\frac {{\mbox{d}}x}{x\ln \!x}}=\ln \!|\!\ln \!x|$

$\int {\frac {{\mbox{d}}x}{x^{n}\ln \!x}}=\ln \!|\!\ln \!x|+\sum _{I=1}^{\infty }(-1)^{I}{\frac {(n-1)^{I}(\ln \!x)^{I}}{I!I}}$

$\int {\frac {{\mbox{d}}x}{x(\ln \!x)^{n}}}={\frac {1}{(1-n)(\ln \!x)^{n-1}}}\qquad (n\neq 1)$

$\int \sin \ln \!x\ {\mbox{d}}x={\frac {x(\sin \ln x-\cos \ln \!x)}{2}}$

$\int \cos \ln \!x\ {\mbox{d}}x={\frac {x(\sin \ln x+\cos \ln \!x)}{2}}$

$\int e^{x}\left(x\ln \!x-x-{\frac {1}{x}}\right)\ {\mbox{d}}x=e^{x}(x\ln \!x-x-\ln \!x)$