Calculus

Integration by Substitution

Change variables to turn a difficult integral into a standard one.

When to use substitution

Use substitution when part of the integrand is almost the derivative of another part. It is the reverse of the chain rule.

\[ \int f(g(x))g'(x)\,dx=\int f(u)\,du \quad\text{where } u=g(x) \]

Worked example 1

Find \(\int 2x(x^2+1)^5\,dx\).

  1. Let \(u=x^2+1\).
  2. Then \(du=2x\,dx\).
  3. The integral becomes \(\int u^5\,du=\frac{u^6}{6}+C\).
  4. Substitute back: \(\frac{(x^2+1)^6}{6}+C\).

Worked example 2: limits

Evaluate \(\int_0^1 2x\sqrt{x^2+4}\,dx\).

  1. Let \(u=x^2+4\), so \(du=2x\,dx\).
  2. Change limits: \(x=0\Rightarrow u=4\), \(x=1\Rightarrow u=5\).
  3. \(\int_4^5 u^{1/2}\,du=\left[\frac{2}{3}u^{3/2}\right]_4^5=\frac{2}{3}(5\sqrt5-8)\).

Tricky substitution for integration

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Substitution method worked example

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Substitution with limits

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