Integration by Parts

Real Analysis · Axiom Academy

The product rule, run in reverse — trading a hard integral for an easier one by choosing what to differentiate. 1. Run the Product Rule Backwards Start from the product rule for derivatives. Integrate both sides , recognize that integrating a derivative undoes it, and rearrange. What falls out is the integration-by-parts formula. Integrate both sides — the left side collapses to uv Picture the formula as a swap. The original integral becomes a finished piece uv minus a leftover integral . You differentiate u to get du and integrate dv to get v — moves that run in opposite directions. Split the integrand into a part to differentiate ( u ) and a part to integrate ( dv ). Differentiate u , integrate dv . No +C on v yet — one constant at the end. Write . The first term is done; the integral is what remains. Solve . A good split makes this one elementary. 3. LIATE: Which Part Becomes u The whole technique hinges on picking u . The LIATE order ranks function types by how good a choice they make for u — take the one that appears earliest on the list, and let everything else be dv . Here x is algebraic and e^x is exponential. LIATE says A beats E, so u = x and . Differentiating x kills it down to a constant — exactly the simplification we want. Algebraic ( x ) outranks exponential ( e^x ), so x is the part we differentiate. Step 4 — the leftover is trivial 5. More Cases — and the Two Clever Tricks

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