List of integrals of logarithmic functions

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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.

## Contents

* 1 Integrals involving only logarithmic functions
* 2 Integrals involving logarithmic and power functions
* 3 Integrals involving logarithmic and trigonometric functions
* 4 Integrals involving logarithmic and exponential functions
* 5 n consecutive integrations
* 6 See also
* 7 References

## Integrals involving only logarithmic functions

∫logaxdx=xlogax−xln⁡a=xln⁡a(ln⁡x−1)

∫ln⁡(ax)dx=xln⁡(ax)−x=x(ln⁡(ax)−1)

∫ln⁡(ax+b)dx=ax+ba(ln⁡(ax+b)−1)

∫(ln⁡x)2dx=x(ln⁡x)2−2xln⁡x+2x

∫lnnxdx=(−1)nx∑k=0nn!k!(−ln⁡x)k

∫dxln⁡x=ln⁡|ln⁡x|+ln⁡x+∑k=2∞(ln⁡x)kk⋅k!

∫dxln⁡x=li⁡(x), the logarithmic integral.

∫dx(ln⁡x)n=−x(n−1)(ln⁡x)n−1+1n−1∫dx(ln⁡x)n−1(for n≠1)

∫ln⁡f(x)dx=xln⁡f(x)−∫xf′(x)f(x)dx(for differentiable f(x)>0)

## Integrals involving logarithmic and power functions

∫xmln⁡xdx=xm+1(ln⁡xm+1−1(m+1)2)(for m≠−1)

∫xm(ln⁡x)ndx=xm+1(ln⁡x)nm+1−nm+1∫xm(ln⁡x)n−1dx(for m≠−1)

∫(ln⁡x)ndxx=(ln⁡x)n+1n+1(for n≠−1)

∫ln⁡xdxxm=−ln⁡x(m−1)xm−1−1(m−1)2xm−1(for m≠1)

∫(ln⁡x)ndxxm=−(ln⁡x)n(m−1)xm−1+nm−1∫(ln⁡x)n−1dxxm(for m≠1)

∫xmdx(ln⁡x)n=−xm+1(n−1)(ln⁡x)n−1+m+1n−1∫xmdx(ln⁡x)n−1(for n≠1)

∫dxxln⁡x=ln⁡|ln⁡x|

∫dxxln⁡xln⁡ln⁡x=ln⁡|ln⁡|ln⁡x||, etc.

∫dxxln⁡ln⁡x=li⁡(ln⁡x)

∫dxxnln⁡x=ln⁡|ln⁡x|+∑k=1∞(−1)k(n−1)k(ln⁡x)kk⋅k!

∫dxx(ln⁡x)n=−1(n−1)(ln⁡x)n−1(for n≠1)

∫ln⁡(x2+a2)dx=xln⁡(x2+a2)−2x+2atan−1xa

∫xx2+a2ln⁡(x2+a2)dx=14ln2(x2+a2)

## Integrals involving logarithmic and trigonometric functions

∫sin⁡(ln⁡x)dx=x2(sin⁡(ln⁡x)−cos⁡(ln⁡x))

∫cos⁡(ln⁡x)dx=x2(sin⁡(ln⁡x)+cos⁡(ln⁡x))

## Integrals involving logarithmic and exponential functions

∫ex(xln⁡x−x−1x)dx=ex(xln⁡x−x−ln⁡x)

∫1ex(1x−ln⁡x)dx=ln⁡xex

∫ex(1ln⁡x−1x(ln⁡x)2)dx=exln⁡x

## n consecutive integrations

For n consecutive integrations, the formula

∫ln⁡xdx=x(ln⁡x−1)+C0

generalizes to

∫⋯∫ln⁡xdx⋯dx=xnn!(ln⁡x−∑k=1n1k)+∑k=0n−1Ckxkk! ,
where Ck are arbitrary constants of integration.

## See also

* List of mathematical identities
* Lists of mathematics topics - None

## References

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

* v
* t
* e

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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