Introduction

The substitution methods of the preceding chapter evaluate many antiderivatives, but an integrand does not always display the derivative of a useful inner function. In this chapter, we develop further methods that rewrite such integrands in forms whose antiderivatives are familiar.

Integration by parts transfers a derivative from one factor in a product to another. Trigonometric identities simplify products and powers of trigonometric functions, and trigonometric substitutions handle square roots of quadratic expressions. Finally, partial fraction decomposition separates a rational function into simpler rational functions.

Choosing a technique is not mechanical. First simplify the integrand, then look for a form that one of the methods below can turn into a known integral. As always, differentiation provides a useful check on an antiderivative.