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

: $\int\log_a x\,dx = x\log_a x - \frac = \frac(\ln x - 1)$

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

: $\int\ln (ax + b)\,dx = \frac(\ln(ax+b) - 1)$

: $\int (\ln x)^2\,dx = x(\ln x)^2 - 2x\ln x + 2x$

: $\int (\ln x)^n\,dx = (-1)^n n! x \sum^_ \frac$

: $\int \frac = \ln, \ln x, + \ln x + \sum^\infty_\frac$

: $\int \frac = \operatorname(x)$, the logarithmic integral.

: $\int \frac = -\frac + \frac\int\frac \qquad\mboxn\neq 1\mbox$

: $\int \ln f(x)\,dx = x\ln f(x) - \int x\frac\,dx \qquad\mbox f(x) > 0\mbox$

Integrals involving logarithmic and power functions

: $\int x^m\ln x\,dx = x^\left(\frac-\frac\right) \qquad\mboxm\neq -1\mbox$

: $\int x^m (\ln x)^n\,dx = \frac - \frac\int x^m (\ln x)^ dx \qquad\mboxm\neq -1\mbox$

: $\int \frac = \frac \qquad\mboxn\neq -1\mbox$

: $\int \frac = -\frac-\frac \qquad\mboxm\neq 1\mbox$

: $\int \frac = -\frac + \frac\int\frac \qquad\mboxm\neq 1\mbox$

: $\int \frac = -\frac + \frac\int\frac \qquad\mboxn\neq 1\mbox$

: $\int \frac = \ln \left, \ln x\$

: $\int \frac = \ln \left, \ln \left, \ln x\ \$, etc.

: $\int \frac = \operatorname(\ln x)$

: $\int \frac = \ln \left, \ln x\ + \sum^\infty_ (-1)^k\frac$

: $\int \frac = -\frac \qquad\mboxn\neq 1\mbox$

: $\int \ln(x^2+a^2)\,dx = x\ln(x^2+a^2)-2x+2a\tan^ \frac$

: $\int \frac\ln(x^2+a^2)\,dx = \frac \ln^2(x^2+a^2)$

Integrals involving logarithmic and trigonometric functions

: $\int \sin (\ln x)\,dx = \frac(\sin (\ln x) - \cos (\ln x))$

: $\int \cos (\ln x)\,dx = \frac(\sin (\ln x) + \cos (\ln x))$

Integrals involving logarithmic and exponential functions

: $\int e^x \left(x \ln x - x - \frac\right)\,dx = e^x (x \ln x - x - \ln x)$

: $\int \frac \left( \frac-\ln x \right)\,dx = \frac$

: $\int e^x \left( \frac- \frac \right)\,dx = \frac$

''n'' consecutive integrations

For $n$ consecutive integrations, the formula

: $\int\ln x\,dx = x(\ln x - 1) +C_$

generalizes to

: $\int\dotsi\int\ln x\,dx\dotsm dx = \frac\left(\ln\,x-\sum_^\frac\right)+ \sum_^ C_ \frac$

See also

*
*

References

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

{{Lists of integrals

Logarithmic functions