List of integrals of logarithmic functions

: $\backslash int\backslash log\_a\; x\backslash ,dx\; =\; x\backslash log\_a\; x\; -\; \backslash frac\{x\}\{\backslash ln\; a\}\; =\; \backslash frac\{x\}\{\backslash ln\; a\}(\backslash ln\; x\; -\; 1)$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

: $\backslash int\; \backslash frac\{dx\}\{x\; \backslash ln\; x\}\; =\; \backslash ln\; \backslash left|\backslash ln\; x\backslash right|$

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

: $\backslash int\; \backslash frac\{dx\}\{x\backslash ln\; \backslash ln\; x\}\; =\; \backslash operatorname\{li\}(\backslash ln\; x)$

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

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

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

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

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

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

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

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

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

For $n$ consecutive integrations, the formula

: $\backslash int\backslash ln\; x\backslash ,dx\; =\; x(\backslash ln\; x\; -\; 1)\; +C\_\{0\}$

generalizes to

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

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{{reflist}}

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. A few integrals are listed on [http://www.math.sfu.ca/~cbm/aands/page_69.htm page 69].

{{Lists of integrals}}

Logarithmic functions