Methods of Integration — Overview

So far every integral has been reduced to a standard form purely by algebraic or trigonometric rewriting. That approach runs out quickly — most integrands met in practice need one of three systematic methods: (1) integration by substitution, which changes the variable so the integral collapses to a known form; (2) integration by parts, the reverse of the product rule, used when the integrand is a product of two dissimilar functions; and (3) integration by partial fractions, used to split a rational function into simpler pieces before integrating. The rest of this chapter develops each method in turn, along with several important families of integrals ($\int\frac{dx}{ax^2+bx+c}$, $\int\frac{dx}{a\sin^2x+b\cos^2x+c}$, $\int\frac{dx}{a\sin x+b\cos x+c}$ …