Selecting Techniques for Antidifferentiation

Integrals that look alike can need totally different moves, because antidifferentiation has no single universal rule, unlike differentiation's chain rule. There's no automatic reverse chain rule that works on every product, since a product rule going backward would require you to already know how to split a function into two pieces whose derivatives behave the right way, and most products don't offer that split cleanly. So instead of a formula, you need a scanning habit.

Scan the integrand for a piece and its derivative. If you spot a function and something that looks like its derivative multiplied alongside it, try u-substitution. If the integrand is a fraction where the numerator's degree is equal to or higher than the denominator's, try polynomial division first to simplify before integrating. If it's a known form already, like 1/(x²+1), recognize it directly as an arctangent, no substitution needed. The decision tree is: rewrite if it simplifies algebraically, divide if it's an awkward rational expression, substitute if a derivative pair is hiding inside.

This topic sits at the end of the antidifferentiation techniques because it's the synthesis step: everything from basic reverse power rule through substitution gets tested together here, with no label telling you which to use. It also sets up later integration by parts and partial fractions, where the same scanning instinct, look at structure before grabbing a method, matters even more.