Choosing an integration technique requires recognizing structural cues in an integrand: a composition of functions multiplied by (a constant times) the derivative of the inner function suggests u-substitution, while a product of a function that simplifies under differentiation (e.g., an inverse trigonometric function, which behaves like a logarithm in this respect) and a function that is easy to antidifferentiate suggests integration by parts. When integration by parts leaves a rational function to integrate, partial fraction decomposition — splitting a rational function with a factored denominator into a sum of simpler terms with unknown coefficients solved via the cover-up method or by clearing denominators — provides a systematic route to a closed-form antiderivative. This belongs to integral calculus, specifically the strategic selection among substitution, integration by parts, and partial fractions as complementary techniques for evaluating both definite and indefinite integrals.
Choosing an integration technique requires recognizing structural cues in an integrand: a composition of functions multiplied by (a constant times) the derivative of the inner function suggests u-sub…