Lecture 6.pdf - LECTURE 6 INTEGRATION BY PART PO-NING CHEN...

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LECTURE 6, INTEGRATION BY PARTPO-NING CHENIn this lecture, we will study the method of integration by part, which is related to theproduct rule of differentiation.1.Product Rule and Integration by PartWe start by recalling the product rule for differentiation. It is a rule for differentiatingthe product of two functions. Namely,(uv)0=u0v+uv0Again, this mean thatuvis an antiderivative of the right hand side and we haveZu0v+uv0dx=uv+cTo carry out integration by part, we usually rewrite the above asZuv0dx=uv-Zu0vdxor simplyZudv=uv-ZvduThe main difficulty in applying integration by substitution is usually finding the correctuandvto use. We will look at different types of examples today.Example 1ComputeZxexdxusing integration by part.Answer to Example 1We chooseu=xdv=exdxWe have thendu=dxv=ex1
2PO-NING CHENandZxexdx=Zudv=uv-Zvdu=xex-Zexdx=xex-ex+cExercise 1ComputeZxsinxdxusing integration by part.

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