u-Substitution Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
Find ∫ln⁡xx dx\displaystyle\int \frac{\ln x}{x}\,dx.

Solution

  1. 1
    Let u=ln⁡xu = \ln x, du=dxxdu = \frac{dx}{x}.
  2. 2
    ∫u du=u22+C=(ln⁡x)22+C\int u\,du = \frac{u^2}{2} + C = \frac{(\ln x)^2}{2} + C.

Answer

(ln⁡x)22+C\frac{(\ln x)^2}{2} + C
Recognising dxx=d(ln⁡x)\frac{dx}{x} = d(\ln x) is the key. After substituting u=ln⁡xu = \ln x, the integral is simply ∫u du\int u\,du.

About u-Substitution

An integration technique where you substitute u=g(x)u = g(x) and du=g′(x) dxdu = g'(x)\,dx to transform a complicated integral into a simpler one. It is the reverse of the chain rule for differentiation.

Learn more about u-Substitution →

More u-Substitution Examples

Example 1 easy

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Example 2 medium

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Example 3 easy

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