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Integration By Parts (9/10/08) Whereas substitution techniques tries (if possible) to reverse the chain rule, “integration by parts” tries to reverse the product rule. Example: x e x dx ?? – Substitution? No! – Question: Can the integrand be split into a product of one part with a nice derivative and another part whose anti-derivative isn’t bad?

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Reversing the product rule If u and v are functions of x, then by the product rule: d/dx (u v) = u v + u v Rewrite: u v = d/dx (u v) - u v Integrate both sides, obtaining the Integration by Parts Formula: u v dx = u v - u v dx The hope, of course, is that u v is easier to integrate than u v was!

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Back to our example x e x dx ?? Well, x ’s derivative is very simple and e x ‘s anti-derivative is no worse, so we try letting u = x and v = e x Then u = 1 and v = e x, so rebuild, using the Parts Formula: x e x dx = x e x - e x dx = x e x – e x + C A quick check, which of course involves the product rule, shows this is right.

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Try some others, and assignment x cos(x) dx ?? x cos(x 2 ) dx ?? x (x+4) dx ?? ln(x) dx ?? (Yes, we can get this one now! Hint: let v = 1) For Friday, read Section 7.1 and do Exercises 1- 15 odd & 19-25 odd

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