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?

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!

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.

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 odd & odd