对数函数积分表

以下是部份对数函數的積分表(書寫時省略了不定積分結果中都含有的任意常數C_n)。

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

$\{\begin{array}{cc}\int \ln x\text{d}x& =x\ln x-\int x\frac{1}{x}\text{d}x\\ & =x\ln x-\int \text{d}x\\ & =x\ln x-x\\ \int \ln (-x)\text{d}x& =-\int \ln (-x)\text{d}(-x)\\ & =-[(-x)\ln (-x)-(-x)]\\ & =x\ln (-x)-x\end{array}⟹\{\begin{array}{cc}\int \ln |x|& =x\ln |x|-x\\ \int {\log }_{\alpha }|x|\text{d}x& =\int \frac{\ln |x|}{\ln \alpha }\text{d}x\\ & =\frac{x\ln |x|-x}{\ln \alpha }\\ & =x{\log }_{\alpha }|x|-\frac{x}{\ln \alpha }\end{array}$

$\int \ln Mx\text{d}x=x\ln Mx-x$

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

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

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

$\int \frac{\text{d}x}{(\ln x{)}^n}=-\frac{x}{(n-1)(\ln x{)}^{n-1}}+\frac{1}{n-1}\int \frac{\text{d}x}{(\ln x{)}^{n-1}}(n\ne 1)$

$\int x^N\ln x\text{d}x=x^{N+1}(\frac{\ln x}{N+1}-\frac{1}{(N+1{)}^2})(N\ne -1)$

$\int x^N(\ln x{)}^n\text{d}x=\frac{x^{N+1}(\ln x{)}^n}{N+1}-\frac{n}{N+1}\int x^N(\ln x{)}^{n-1}dx(N\ne -1)$

$\int \frac{(\ln x{)}^n\text{d}x}{x}=\frac{(\ln x{)}^{n+1}}{n+1}(n\ne -1)$

$\int \frac{\ln x\text{d}x}{x^r}=-\frac{\ln x}{(r-1)x^{r-1}}-\frac{1}{(1-r{)}^2{x}^{r-1}}(r\ne 1)$

$\int \frac{(\ln x{)}^n\text{d}x}{x^r}=-\frac{(\ln x{)}^n}{(r-1)x^{r-1}}+\frac{n}{r-1}\int \frac{(\ln x{)}^{n-1}dx}{x^m}(m\ne 1)$

$\int \frac{x^N\text{d}x}{(\ln x{)}^n}=-\frac{x^{N+1}}{(n-1)(\ln x{)}^{n-1}}+\frac{N+1}{n-1}\int \frac{x^N\text{d}x}{(\ln x{)}^{n-1}}(n\ne 1)$

$\int \frac{\text{d}x}{x\ln x}=\ln |\ln x|$

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

$\int \frac{\text{d}x}{x(\ln x{)}^n}=\frac{1}{(1-n)(\ln x{)}^{n-1}}(n\ne 1)$

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

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

$\int e^x(x\ln x-x-\frac{1}{x})\text{d}x=e^x(x\ln x-x-\ln x)$