U-Substitution

AP Calculus · Axiom Academy

The chain rule in reverse — the most important integration technique Many integrals don't match basic formulas directly. U-substitution simplifies integrals by recognizing a composite function pattern (the chain rule in reverse). Choose u = g(x) (the "inner function") Rewrite the integral entirely in terms of u and du Substitute back to get the answer in terms of x Look for a function and its derivative appearing together Choose u as the "inner" part of a composition Everything must convert to u and du — no leftover x 's For definite integrals, change the bounds when substituting Always verify by differentiating your answer

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