The Parts Formula

Calculus 2 · Axiom Academy

Reverse the product rule to integrate a product — split it into u and dv , then trade for the easier . 1. The Product Rule, Run Backwards Integration by parts is not a new rule — it is the product rule for derivatives , integrated and rearranged. Watch the product rule get integrated on both sides, then solved for the integral we actually want. Integrate both sides (the integral of a derivative is the function itself), then move one integral across: Parts only helps if the new integral is simpler — so you must pick u and dv wisely. The mnemonic LIATE ranks function types; whichever appears earliest in the list becomes u (the part you differentiate), and the rest becomes dv . The integrand is x (Algebraic) times (Trigonometric). A comes before T in LIATE, so u = x and . Watch the four pieces get assembled into . 4. The Whole Strategy at a Glance Every parts problem is the same four-move loop. Watch the integrand flow through it — split, differentiate & integrate, then assemble — landing on the formula. Pick u (earliest in LIATE) and let the rest be dv . and — fill in all four pieces. Evaluate — chosen to be the easier one. You can now integrate a product by reversing the product rule — split with LIATE, then trade for the easier . Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Calculus 2 course.