Radford Mathematics IB Mathematics resources • AA HL

IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Integration by Substitution

Recognise an inner function and its derivative, then change variable to simplify the integral.

AA HL · SL 5.10

Learning goal

Use reverse chain rule and substitution for expressions of the syllabus form.

Syllabus link

AA HL: SL 5.10 · Topic 5 Calculus.

Big idea

Substitution is a change of variable. A complicated integral becomes standard when the integrand contains a function and a matching derivative factor.

Key relationship

\(u=g(x),\quad du=g\'(x)dx\)

1

Core idea

Substitution is a change of variable. A complicated integral becomes standard when the integrand contains a function and a matching derivative factor.

Use reverse chain rule and substitution for expressions of the syllabus form.

2

Key results and method

\[u=g(x),\quad du=g\'(x)dx\]
\[\int k g\'(x)f(g(x))dx\]
  1. Choose \(u\) as the inner expression.
  2. Differentiate to obtain \(du\).
  3. Rewrite the entire integral in \(u\).
  4. Integrate, then substitute back.
3

Worked example

Question

Evaluate \(\int 2x(x^2+1)^4dx\).

Solution
  1. Let \(u=x^2+1\), so \(du=2x\,dx\).
  2. The integral becomes \(\int u^4du\).
Answer: \(\frac15(x^2+1)^5+C\).
4

Practice

Try these questions before moving on.

  1. Evaluate \(\int 4x\sin(x^2)dx\).
  2. Use substitution for a logarithmic-looking quotient.
  3. Evaluate a definite integral by changing the limits or substituting back.

Search the worksheet library for more practice →

5

Tutorials and premium resources

▶ U-substitution — Part 1 ▶ U-substitution — Part 2
Premium printable lesson notes

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