U-Substitution — Integration's Chain Rule

The reverse of the chain rule for derivatives.

Method:

1. Identify an inner function — call it u

2. Compute du = u'dx

3. Rewrite the integral entirely in terms of u and du

4. Integrate

5. Substitute back

Example: ∫2x · cos(x²) dx

Let u = x². Then du = 2x dx.

∫cos(u) du = sin(u) + C = sin(x²) + C

For definite integrals: Change the limits! If u = x² and x goes from 0 to 3, then u goes from 0 to 9.

Spotting u-sub: Look for a function AND its derivative appearing in the integral. The derivative (possibly off by a constant) signals u-sub.

This is the #1 integration technique on the AP exam.

Reference:

Wikipedia: U-Substitution

https://en.wikipedia.org/wiki/Integration_by_substitution