6.9 Integration Using Substitution

Reverse chain rule

u-substitution reverses the chain rule. Let u = the inner function, compute du = u'(x) dx, rewrite the integral in terms of u, integrate, then back-substitute.

When & definite

It works when the integrand contains a function and (a multiple of) its derivative. For a definite integral, change the limits to u-values (or back-substitute before evaluating). Example: ∫2x(x²+1)³ dx uses u = x² + 1.

Key terms

u-substitution

Reverses the chain rule.

Choose u

The inner function; then du = u'(x) dx.

When to use

Integrand has a function and its derivative.

Definite integrals

Change the limits to u-values.

Example

∫2x(x²+1)³ dx: u = x² + 1.

Try an AP-style question

∫ 2x(x² + 1)⁵ dx = ?

1. (x² + 1)⁶/6 + C
2. (x² + 1)⁶ + C
3. 5(x² + 1)⁴ + C
4. x²(x² + 1)⁶/6 + C