How to solve an integral by parts?

Use integration by parts to find the integral of x^2 cos 3x dx how do I solve this?

• Answer:

  Integration by parts: (u v )' = u ' v + u v ' Therefore: Integral of (u' v) = u v - Integral of (u v ') so if you have a product, you can integrate one term and differentiate the other, and you might have a simpler sum. In your case u ' can be cos(3x) because that doesn't really get any more complicated if you integrate it. And then v is x^2, which gets simpler if you differentiate it. integral of x^2 cos 3x dx = (1/3) x^2 sin 3x - (2/3) integral of x sin 3x dx and the new integral is simpler than the old. In fact, you can integrate it using ... integration by parts!