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

\int \log _{a}x\;dx={\frac {x\ln x-x}{\ln a}}

\int\ln ax\;dx = x\ln ax - x

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

\int (\ln x)^2\; dx = x(\ln x)^2 - 2x\ln x + 2x

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

\int \frac{dx}{\ln x} = \ln|\ln x| + \ln x + \sum^\infty_{k=2}\frac{(\ln x)^k}{k\cdot k!}

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

= the logarithmic integral (asymptotically,

{\text{li}}(x)=\Theta ({\frac {x}{\ln x}})

\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{)}

\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{)}

\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{)}

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

\int \frac{\ln{x^n}\;dx}{x} = \frac{(\ln{x^n})^2}{2n} \qquad\mbox{(for }n\neq 0\mbox{)}

\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{)}

\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{)}

\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{)}

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

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

, etc.

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

\int \frac{dx}{x^n\ln x} = \ln \left|\ln x\right| + \sum^\infty_{k=1} (-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}} \qquad\mbox{(for }n\neq 1\mbox{)}

\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)

\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))

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

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

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

For $n$ consecutive integrations, the formula

\int\ln x\;dx = x\;(\ln x - 1) +C_{0}

generalizes to

\int\cdot\cdot\cdot\int\ln x\;dx\cdot\cdot\cdot\;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!}

Lists of integrals

Rational functions Irrational functions Trigonometric functions Inverse trigonometric functions Hyperbolic functions Inverse hyperbolic functions Exponential functions Logarithmic functions Gaussian functions

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