u-substitution

u-substitution rewrites an integral in terms of a new variable so that a familiar antiderivative rule applies. You choose an inner expression and set u equal to it, compute du = u′ dx, then replace every x in the integral with the corresponding u expression. If the substitution is well chosen, the entire integrand — including dx — converts cleanly, and the remaining integral is one you already know how to evaluate.

A worked example makes the pattern clear. For ∫ 2x(x² + 1)⁵ dx, let u = x² + 1, so du = 2x dx. The integral becomes ∫ u⁵ du = u⁶/6 + C, and substituting back gives (x² + 1)⁶/6 + C. The reason it works is the chain rule read backwards: differentiating (x² + 1)⁶/6 produces exactly the original integrand.

The usual choice for u is an inner function whose derivative already appears in the integrand, up to a constant factor — the expression under a radical, inside parentheses raised to a power, in a denominator, or in an exponent. Constants are easy to fix: if du = 2x dx but the integrand only has x dx, write x dx = du/2. Two habits prevent most errors. Convert dx completely, leaving no stray x behind, and with a definite integral either change the limits to u-values or substitute back to x before evaluating — never plug x-limits into a u-expression.

AP Calculus AB introduces u-substitution as the first general integration technique and tests it heavily on both multiple-choice and free-response questions, including definite integrals where the limits must be converted. Recognizing composite structure is the same skill the SAT rewards when you simplify algebraic systems by substituting one expression for another.