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.
Integrals involving only logarithmic functions
![{\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}={\frac {x}{\ln a}(\ln x-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98ef469b919dddd39f50ebf313ba908841f2868c)
![{\displaystyle \int \ln(ax)\,dx=x\ln(ax)-x=x(\ln(ax)-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d65f45006976c4465363e7ed303b4adbbf4e881)
![{\displaystyle \int \ln(ax+b)\,dx={\frac {ax+b}{a}(\ln(ax+b)-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/797d98386b914e19585d6851600f1bf501224d9c)
![{\displaystyle \int (\ln x)^{2}\,dx=x(\ln x)^{2}-2x\ln x+2x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5853a2d856005e03dbeb45c5e1316ad96987fec)
![{\displaystyle \int (\ln x)^{n}\,dx=x\sum _{k=0}^{n}(-1)^{n-k}{\frac {n!}{k!}(\ln x)^{k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46f4ac1cb2650cedf436b2bf6ac167e46781461d)
![{\displaystyle \int {\frac {dx}{\ln x}=\ln |\ln x|+\ln x+\sum _{k=2}^{\infty }{\frac {(\ln x)^{k}{k\cdot k!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b80db5839b7b4074c717480feeac2550240c486)
, 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{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99946b6234c515f8d01d55eb7e0e6cf80df60a7e)
![{\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{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36003856e0f51bde0eda0e7e16f6b1f7657d2e8c)
Integrals involving logarithmic and power functions
![{\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{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9dde18dfe4f0da0ecda646e2803b65d18f8741c)
![{\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{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d907961eeade3559fc4d69fad5c325e846e3ccc)
![{\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x}={\frac {(\ln x)^{n+1}{n+1}\qquad {\mbox{(for }n\neq -1{\mbox{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f58c4cc27d4323a84868e2ad79da5ef9c29f886)
![{\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{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae388283fc663d6b2e6a67c4091f6ea9aa9270f0)
![{\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{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a041742c5699737601a6ff764a8afd472dd4b1c8)
![{\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{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0e9cfdabfcfc28ea149b1e434a1224b8ee0644)
![{\displaystyle \int {\frac {dx}{x\ln x}=\ln \left|\ln x\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fce91dfa19693fbe5883d150963bed432e06d047)
, etc.
![{\displaystyle \int {\frac {dx}{x\ln \ln x}=\operatorname {li} (\ln x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/180e391e734340059024d58f0396fd1c69fca77d)
![{\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!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/500ec1ec23b97b79baf39b8febbb589f11e37b82)
![{\displaystyle \int {\frac {dx}{x(\ln x)^{n}=-{\frac {1}{(n-1)(\ln x)^{n-1}\qquad {\mbox{(for }n\neq 1{\mbox{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acd7a092992c6886fb764bc8b3f5b44ba076fa75)
![{\displaystyle \int \ln(x^{2}+a^{2})\,dx=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb444c640d10b7d12aec5cabd50c55c0a957b820)
![{\displaystyle \int {\frac {x}{x^{2}+a^{2}\ln(x^{2}+a^{2})\,dx={\frac {1}{4}\ln ^{2}(x^{2}+a^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8473d4e4b11e413ddb0d6965f0d0adfe7a58015e)
Integrals involving logarithmic and trigonometric functions
![{\displaystyle \int \sin(\ln x)\,dx={\frac {x}{2}(\sin(\ln x)-\cos(\ln x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33c6c667b8cfe2a07263f42d1392cc2dfdd65cc5)
![{\displaystyle \int \cos(\ln x)\,dx={\frac {x}{2}(\sin(\ln x)+\cos(\ln x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/798da7bc36fe2c1edd580c90342c1cde8086df4d)
Integrals involving logarithmic and exponential functions
![{\displaystyle \int e^{x}\left(x\ln x-x-{\frac {1}{x}\right)\,dx=e^{x}(x\ln x-x-\ln x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8632f3aea3bdd5b6c25fa45547d2d3359960dfbc)
![{\displaystyle \int {\frac {1}{e^{x}\left({\frac {1}{x}-\ln x\right)\,dx={\frac {\ln x}{e^{x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/617a944320af8e067a09c96720fa8ef436f4a69f)
![{\displaystyle \int e^{x}\left({\frac {1}{\ln x}-{\frac {1}{x(\ln x)^{2}\right)\,dx={\frac {e^{x}{\ln x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/021b378a21d01e5131cc4fa11466d9ada8990f58)
n consecutive integrations
For
consecutive integrations, the formula
![{\displaystyle \int \ln x\,dx=x(\ln x-1)+C_{0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45ced9abfbe033de1dab09c1dfa704755885c317)
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!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c34d7df20d2d235e0165f209c08efc563371bb3)
See also
References