How Do You Integrate by Inspection or Substitution?

Use integration by inspection when you can immediately recognize an integrand as the reverse of the chain rule. Use substitution when introducing a new variable makes that same reverse-chain structure clearer.

This is AHL 5.11 content for IB Mathematics: Applications and Interpretation HL and may appear on any GDC-active HL paper.

The choice does not depend on whether the integral is indefinite or definite; it depends on how clearly the composite function and derivative factor can be recognized. Inspection is faster, while substitution makes the reasoning explicit and reduces algebraic mistakes.

The Underlying Mechanism

Both methods use the reverse chain rule:

$$\int f(g(x))g'(x)\,dx=F(g(x))+C,$$