6.9 Integrating Using Substitution

The idea behind substitution

U-substitution reverses the chain rule. If an integral contains a function and its derivative (up to a constant), let u = g(x), so du = g'(x)dx. Rewrite the entire integral in terms of u, integrate, then substitute back for x. This works because it simplifies composite functions into basic forms you already know how to integrate.

Choosing u wisely

Pick u as the 'inside' function of a composition, often what's raised to a power, inside a root, or in an exponent/denominator. Check that du (or a constant multiple) appears elsewhere in the integrand. If it doesn't appear cleanly, you may need to solve for x-terms or try a different u.

Substitution with definite integrals

For definite integrals, either substitute back to x before evaluating, or change the bounds: plug the original x-limits into u = g(x) to get new u-limits, then evaluate directly in terms of u without converting back. Both methods give the same numeric answer.

Key terms

U-substitution

A technique that reverses the chain rule by setting u = g(x) to simplify an integral into a basic form.

How to choose u

Pick u as the inner function of a composition whose derivative (or a constant multiple) also appears in the integrand.

Steps for substitution

1) Let u = g(x). 2) Find du = g'(x)dx. 3) Rewrite integral in terms of u. 4) Integrate. 5) Substitute back for x (or use new bounds).

Changing bounds in definite integrals

Plug the original x-limits into u = g(x) to get new u-limits, then evaluate directly without converting back to x.

∫2x·cos(x²)dx

Let u = x², du = 2x dx → ∫cos(u)du = sin(u) + C = sin(x²) + C.

Try an AP-style question

∫ 2x(x² + 1)⁵ dx = ?

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