Integration by parts formula lagrange notation

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Integration by parts formula (Lagrange notation)integraldisplayf(x)g(x) dx=f(x)G(x)integraldisplayfprime(x)G(x) dx.HereGis an antiderivative ofg.The integration by parts formula is useful for integrating some products ofthe formf(x)g(x). You can see that it changes the problem of integratingf(x)g(x) into the problem of integratingfprime(x)G(x), whereGis anantiderivative ofg.So it’s useful for integrating products of the formf(x)g(x) that have thefollowing characteristics.You can find an antiderivativeGofg; in other words, you canintegrate the expressiong(x).The productfprime(x)G(x) is easier to integrate than the productf(x)g(x). For example, this may be the case iffprime(x) is simpler thanf(x).When we integrate by using the integration by parts formula, we say thatwe’reintegrating by parts. Here’s an example.177
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
4Integration by partsby parts in fairly simple cases, like the ones that you’ve seen so far, whichis usually all that you need to do.Integration by parts formula (informal)integral of product = (first)×parenleftbiggantiderivativeof secondparenrightbiggintegral ofparenleftbiggparenleftbiggderivativeof firstparenrightbigg×parenleftbiggantiderivativeof secondparenrightbiggparenrightbigg.

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