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Integral calculus
Lesson 13 of 75
Standard practice

Integration by substitution

Objective. Transform integrals with justified changes of variable.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Integration by substitution reverses the chain rule: if u=g(x), then du=g'(x)dx, allowing ∫f(g(x))g'(x)dx to be rewritten as ∫f(u)du in the new variable, which is often easier to integrate. The change of variable must be justified by correctly transforming the differential and, for definite integrals, the limits of integration into u-values. For ∫2x·cos(x²)dx, set u=x², so du=2x dx, giving ∫cos(u)du=sin(u)+C=sin(x²)+C. Choosing u as the "inner function" whose derivative (up to a constant) also appears in the integrand is the key recognition skill.

Worked example

Evaluate ∫2x·cos(x²)dx using integration by substitution.

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Illustrated card showing 2
  1. 1Let u=x², so du=2x dx, meaning 2x dx can be replaced directly by du.
  2. 2Rewrite the integral entirely in u: ∫cos(u)du.
  3. 3Integrate: ∫cos(u)du = sin(u) + C.
  4. 4Substitute back u=x² to give the final answer sin(x²) + C.

Answer: sin(x²) + C

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Common mistake. Students substitute u=g(x) but forget to replace dx with du/g'(x), instead leaving a leftover x term unconverted into u, producing an integral that mixes two variables and cannot be evaluated.

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