# partsexample - An example of integration by parts We want...

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An example of integration by parts.We want to find the antiderivative ofx5ex2dx.There is more than one way to do this. The point of this handout is to try two differentways, and to practice combining substitution with integration by parts.The first method is to use substitution to make the integral easier, and then use inte-gration by parts.The second is to use integration by parts directly. Here it might be a little harder tosee how to choose the parts.1. Substitution, then integration by parts.Starting withu=x2, we computedu= 2x dx. Solving fordxgivesdx=du2x.If wesubstitute the formula fordxinto the original integral, we getx5ex2du2x=12x4ex2du.We now want to write everything leftover in terms ofu. Sinceu=x2, we can writeex2aseu, andx4asu2. So, after substitution, the integral becomes12u2eudu.This looks a lot like the integrals we’ve been considering, and seems a good candidatefor integration by parts.Depending on how you like to remember integration by parts, you might run intoa little notational problem trying to solve the integral above.If you like theu dvmethod, then it’s going to be a bit awkward to apply – there’s already a variablecalled “u