Integrationby partsA special rule,integration by parts, is available for integrating products of two functions.This unit derives and illustrates this rule with a number of examples.In order to master the techniques explained here it is vital that you undertake plenty of practiceexercises so that they become second nature.After reading this text, and/or viewing the video tutorial on this topic, you should be able to:•state the formula for integration by parts•integrate products of functions using integration by partsContents1.Introduction22.Derivation of the formula for integration by partsintegraldisplayudvdxdx=u v−integraldisplayvdudxdx23.Using the formula for integration by parts51ccirclecopyrtmathcentre July 20, 2005
1. IntroductionFunctions often arise as products of other functions, and we may be required to integrate theseproducts. For example, we may be asked to determineintegraldisplayxcosxdx .Here, the integrand is the product of the functionsxand cosx. A rule exists for integratingproducts of functions and in the following section we will derive it.2. Derivation of the formula for integration by partsWe already know how to differentiate a product: ify=u vthendydx=d(uv)dx=udvdx+vdudx.Rearranging this rule:udvdx=d(uv)dx−vdudx.Now integrate both sides:integraldisplayudvdxdx=integraldisplayd(uv)dxdx−integraldisplayvdudxdx .The first term on the right simplifies since we are simply integrating what has been differentiated.integraldisplayudvdxdx=u v−integraldisplayvdudxdx .This is the formula known asintegration by parts.Key PointIntegration by partsintegraldisplayudvdxdx=u v−integraldisplayvdudxdxThe formula replaces one integral (that on the left) with another (that on the right); the intentionis that the one on the right is a simpler integral to evaluate, as we shall see in the followingexamples.