web-int-byparts-dec03 - Integration by parts A special rule integration by parts is available for integrating products of two functions This unit

# web-int-byparts-dec03 - Integration by parts A special rule...

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Integration by parts A 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 practice exercises 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 parts Contents 1. Introduction 2 2. Derivation of the formula for integration by parts integraldisplay u d v d x d x = u v integraldisplay v d u d x d x 2 3. Using the formula for integration by parts 5 1 c circlecopyrt math centre July 20, 2005
1. Introduction Functions often arise as products of other functions, and we may be required to integrate these products. For example, we may be asked to determine integraldisplay x cos x d x . Here, the integrand is the product of the functions x and cos x . A rule exists for integrating products of functions and in the following section we will derive it. 2. Derivation of the formula for integration by parts We already know how to differentiate a product: if y = u v then d y d x = d( uv ) d x = u d v d x + v d u d x . Rearranging this rule: u d v d x = d( uv ) d x v d u d x . Now integrate both sides: integraldisplay u d v d x d x = integraldisplay d( uv ) d x d x integraldisplay v d u d x d x . The first term on the right simplifies since we are simply integrating what has been differentiated. integraldisplay u d v d x d x = u v integraldisplay v d u d x d x . This is the formula known as integration by parts . Key Point Integration by parts integraldisplay u d v d x d x = u v integraldisplay v d u d x d x The formula replaces one integral (that on the left) with another (that on the right); the intention is that the one on the right is a simpler integral to evaluate, as we shall see in the following examples.