List of integrals of logarithmic functions

Template:Short description The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals.

Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.

Integrals involving only logarithmic functions

$\int {\log }_{a}xdx=x{\log }_{a}x-\frac{x}{\ln a}=\frac{x}{\ln a}(\ln x-1)$

$\int \ln (ax)dx=x\ln (ax)-x=x(\ln (ax)-1)$

$\int \ln (ax+b)dx=\frac{ax+b}{a}(\ln (ax+b)-1)$

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

$\int {\ln }^{n}xdx=(-1{)}^{n}x\sum _{k=0}^n\frac{n!}{k!}(-\ln x{)}^k$

$\int \frac{dx}{\ln x}=\ln |\ln x|+\ln x+\sum _{k=2}^{\mathrm{\infty }}\frac{(\ln x{)}^k}{k\cdot k!}$

$\int \frac{dx}{\ln x}=\mathrm{li}(x)$, the logarithmic integral.

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

$\int \ln f(x)dx=x\ln f(x)-\int x\frac{f^{\prime }(x)}{f(x)}dx\text{(for differentiable }f(x)>0\text{)}$

Integrals involving logarithmic and power functions

$\int x^m\ln xdx=x^{m+1}(\frac{\ln x}{m+1}-\frac{1}{(m+1{)}^2})\text{(for }m\ne -1\text{)}$

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

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

$\int \frac{\ln xdx}{x^m}=-\frac{\ln x}{(m-1)x^{m-1}}-\frac{1}{(m-1{)}^2{x}^{m-1}}\text{(for }m\ne 1\text{)}$

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

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

$\int \frac{dx}{x\ln x}=\ln |\ln x|$

$\int \frac{dx}{x\ln x\ln \ln x}=\ln |\ln |\ln x||$, etc.

$\int \frac{dx}{x\ln \ln x}=\mathrm{li}(\ln x)$

$\int \frac{dx}{x^n\ln x}=\ln |\ln x|+\sum _{k=1}^{\mathrm{\infty }}(-1{)}^k\frac{(n-1{)}^k(\ln x{)}^k}{k\cdot k!}$

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

$\int \ln (x^2+a^2)dx=x\ln (x^2+a^2)-2x+2a{\tan }^{-1}\frac{x}{a}$

$\int \frac{x}{x^2+a^2}\ln (x^2+a^2)dx=\frac{1}{4}{\ln }^2(x^2+a^2)$

Integrals involving logarithmic and trigonometric functions

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

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

Integrals involving logarithmic and exponential functions

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

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

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

n consecutive integrations

For $n$ consecutive integrations, the formula

$\int \ln xdx=x(\ln x-1)+C_0$

generalizes to

$\int \cdots \int \ln xdx\cdots dx=\frac{x^n}{n!}(\ln x-\sum _{k=1}^n\frac{1}{k})+\sum _{k=0}^{n-1}{C}_k\frac{x^k}{k!}$ ,

where $C_k$ are arbitrary constants of integration.

See also

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References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. A few integrals are listed on page 69.