List of integrals of logarithmic functions

Note: <math>x > 0</math> is assumed throughout this article.

<math>\int\ln x\,dx = x\ln x - x</math>

<math>\int (\ln x)^2 dx = x(\ln x)^2 - 2x\ln x + 2x</math>

<math>\int (\ln x)^n dx = x(\ln x)^n - n\int (\ln x)^{n-1} dx \qquad\mbox{(for }n\neq 1\mbox{)}</math>

<math>\int \frac{dx}{\ln x} = \ln|\ln x| + \ln x + \sum^\infty_{i=2}\frac{(\ln x)^i}{i\cdot i!}</math>

<math>\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{)}</math>

<math>\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{)}</math>

<math>\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,n\neq 1\mbox{)}</math>

<math>\int \frac{(\ln x)^n dx}{x} = \frac{(\ln x)^{n+1}}{n+1} \qquad\mbox{(for }n\neq 1\mbox{)}</math>

<math>\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{)}</math>

<math>\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,n\neq 1\mbox{)}</math>

<math>\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{)}</math>

<math>\int \frac{dx}{x\ln x} = \ln|\ln x|</math>

<math>\int \frac{dx}{x^n\ln x} = \ln|\ln x| + \sum^\infty_{i=1} (-1)^i\frac{(n-1)^i(\ln x)^i}{i\cdot i!}</math>

<math>\int \frac{dx}{x (\ln x)^n} = -\frac{1}{(n-1)(\ln x)^{n-1}} \qquad\mbox{(for }n\neq 1\mbox{)}</math>

<math>\int \sin (\ln x)\,dx = \frac{x}{2}(\sin (\ln x) - \cos (\ln x))</math>

<math>\int \cos (\ln x)\,dx = \frac{x}{2}(\sin (\ln x) + \cos (\ln x))</math>