Integration by substitution
Reverse chain rule. Replace a function and its derivative with u and du.
Y13. Look for a function and its derivative in the integrand. Substitute u = f(x), du = f'(x)dx.
Worked examples
∫ 2x cos(x²) dx: u = x², du = 2x dx ⇒ ∫ cos u du = sin u + c = sin(x²) + c.
∫ x/(x²+1) dx: u = x²+1 ⇒ ∫ (1/2)/u du = ½ ln|u| + c.
Definite: change limits too: ∫ab → ∫u(a)u(b).
Frequently asked questions
How to spot a substitution?
Look for a function and its derivative (up to a constant multiple) in the integrand.
Definite integrals?
Either change the limits to u-values OR substitute back to x and use original limits.