U-Substitution Integration

Calculus 1 · Axiom Academy

EXAMPLE U-Substitution Integration Working an integral by reversing the chain rule — spot the inner function, substitute, and integrate. Evaluate the indefinite integral . The exponential is composed with x^3 , and an x^2 sits out front — a classic signal for the substitution u = x^3 . Nice work — you reversed the chain rule by hand. The moves worth keeping: Pick u as the inner function: here u = x^3 , the thing tucked inside the exponential. A good choice makes its derivative appear (up to a constant) elsewhere in the integrand. Differentiate, then solve for what you have: , so . The leftover is exactly what's sitting in the integral. The constant rides along: the pulls out front; . Integrate in u , then undo the substitution: . Check by differentiating: — back to the integrand. Substitution is the workhorse of integration: whenever you see a function and (a constant times) its derivative together, let u be that inner function and the integral collapses.

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