Unit 8Integration methodsExample 23Integrating by partsFind the integralintegraldisplayxsinxdx.SolutionThe integrand is a product of two expressions,xand sinx. Youcan integrate the second expression, sinx, and differentiating the firstexpression,x, makes it simpler. So try integration by parts.Letf(x) =xandg(x) = sinx.Thenfprime(x) = 1, and an antiderivative ofg(x) isG(x) =−cosx.So the integration by parts formula givesintegraldisplayxsinxdx=integraldisplayf(x)g(x) dx=f(x)G(x)−integraldisplayfprime(x)G(x) dx=x×(−cosx)−integraldisplay1×(−cosx) dx=−xcosx+integraldisplaycosxdxFind the integral in this expression.=−xcosx+ sinx+c.You can see that in Example 23 the use of the integration by parts formulachanged the problem of integratingxsinxinto the problem of integratingcosx, which is easier. Here are two similar examples for you to try.Activity 34Integrating by partsFind the following integrals.(a)integraldisplayxcosxdx(b)integraldisplayxexdxAs you become more familiar with integration by parts, you’ll probablyfind that you can carry it out more quickly and conveniently if youremember the informal version of the formula stated below. You can recitethis version in your head as you apply integration by parts, in a similarway to the informal versions of the product and quotient rules fordifferentiation that you met in Unit 7. It’s useful for applying integration178