List of integrals of logarithmic functions

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.

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

${\displaystyle \int \ln(ax)\,dx=x\ln(ax)-x=x(\ln(ax)-1)}$

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

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

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

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

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

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

${\displaystyle \int \ln f(x)\,dx=x\ln f(x)-\int x{\frac {f'(x)}{f(x)}\,dx\qquad {\mbox{(for differentiable }f(x)>0{\mbox{)}$

${\displaystyle \int x^{m}\ln x\,dx=x^{m+1}\left({\frac {\ln x}{m+1}-{\frac {1}{(m+1)^{2}\right)\qquad {\mbox{(for }m\neq -1{\mbox{)}$

${\displaystyle \int x^{m}(\ln x)^{n}\,dx={\frac {x^{m+1}(\ln x)^{n}{m+1}-{\frac {n}{m+1}\int x^{m}(\ln x)^{n-1}dx\qquad {\mbox{(for }m\neq -1{\mbox{)}$

${\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x}={\frac {(\ln x)^{n+1}{n+1}\qquad {\mbox{(for }n\neq -1{\mbox{)}$

${\displaystyle \int {\frac {\ln x\,dx}{x^{m}=-{\frac {\ln x}{(m-1)x^{m-1}-{\frac {1}{(m-1)^{2}x^{m-1}\qquad {\mbox{(for }m\neq 1{\mbox{)}$

${\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x^{m}=-{\frac {(\ln x)^{n}{(m-1)x^{m-1}+{\frac {n}{m-1}\int {\frac {(\ln x)^{n-1}dx}{x^{m}\qquad {\mbox{(for }m\neq 1{\mbox{)}$

${\displaystyle \int {\frac {x^{m}\,dx}{(\ln x)^{n}=-{\frac {x^{m+1}{(n-1)(\ln x)^{n-1}+{\frac {m+1}{n-1}\int {\frac {x^{m}dx}{(\ln x)^{n-1}\qquad {\mbox{(for }n\neq 1{\mbox{)}$

${\displaystyle \int {\frac {dx}{x\ln x}=\ln \left|\ln x\right|}$

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

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

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

${\displaystyle \int {\frac {dx}{x(\ln x)^{n}=-{\frac {1}{(n-1)(\ln x)^{n-1}\qquad {\mbox{(for }n\neq 1{\mbox{)}$

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

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

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

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

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

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

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

For ${\displaystyle n}$ consecutive integrations, the formula

${\displaystyle \int \ln x\,dx=x(\ln x-1)+C_{0}$

generalizes to

${\displaystyle \int \dotsi \int \ln x\,dx\dotsm dx={\frac {x^{n}{n!}\left(\ln \,x-\sum _{k=1}^{n}{\frac {1}{k}\right)+\sum _{k=0}^{n-1}C_{k}{\frac {x^{k}{k!}$

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• 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.