Lecture 7.pdf - LECTURE 8 MORE ON INTEGRATION BY PART...

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LECTURE 8, MORE ON INTEGRATION BY PART.PO-NING CHENIn this lecture, we will more complicated examples of integration. In these examples, wehave to apply the methods we learned before multiple times.1.Integration by Substitution and Integration by PartRecall the two methods we learn from the last three lectures:First we have the integration by substitution:Zf0(g(x))g0(x)dx=f(g(x)) +c.We also learned about integration by part:Zudv=uv-ZvduThe main difficulty in applying these methods is usually finding the correctuand/orvtouse. We will look at different types of examples today.2.Integration by Part TwiceExample 1ComputeZexsinxdx.Answer to Example 1We chooseu=exdv= sinxdxWe have thendu=exdxv=-cosxandZexsinxdx=Zudv=uv-Zvdu=-excosx+ZexcosxdxFrom here we perform a second integration by part. We chooseu=exdv= cosxdx1
2PO-NING CHENWe have thendu=exdxv= sinxandZexcosxdx=Zudv=uv-Zvdu=exsinx-ZexsinxdxCombining the two, we getZexsinxdx=e

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