Integration by Substitution

Real Analysis · Axiom Academy

LESSON Integration by Substitution The reverse chain rule: spot an inner function whose derivative is already there, swap to u , and the integral collapses. 1. The Connection to the Chain Rule Differentiating a composite with the chain rule gives a product: the derivative of the outer function evaluated at the inner one, times the derivative of the inner one. Substitution is that arrow reversed . The move is mechanical once you see the pattern. Name the inner function u , differentiate it to get du , and replace every x -piece so nothing in x remains. 3. Example: A Linear Inner Function 4. Example: When the Constant Is Off 5. Definite Integrals: Move the Limits For a definite integral you have a choice. You can back-substitute and use the original x -bounds — or, cleaner, convert the bounds to u -values once and never return to x . If u=g(x) , then x=a becomes u=g(a) and x=b becomes u=g(b) . You can now spot the chain-rule pattern inside an integral, collapse it with u=g(x) , and handle the constant and the limits. Scroll up to revisit any step.

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