Efficient Integration by Parts: Solving \int x^2*cosx dx with Step-by-Step Guide

[Aerosion]

Homework Statement

$$\int x^2*cosx dx$$

Homework Equations

The Attempt at a Solution

Okay, so I started by making...
u=x^2
du=2x
dv=cos(x)
v=-sin(x)

Then I made the rudimentary equation:

$$\int x^2 * cos(x) dx = -x^2*sin(x) + \int 2x * sin(x) dx$$

Then I took the last integration problem (the one all the way on the right) and did integration by parts on that one again, to make:

u=2x
du=2
dv=sin(x)
v=-cos(x)

to make...

$$\int x^2 * cos(x) dx = -x^2 * sin(x) + 2x * sin(x) + \int 2 * cos(x) dx$$

I evaluated the last integration problem to be -2*sin(x), and got a final answer of sin(x)(-x^2+2x-2) (after factoring). Of course, my calculator says this is wrong, so where'd I mess up?

Thanks. (I'm getting better with Latex, btw)

 

[cristo]

In your first step, if dv=cos(x)dx, then v=sin(x) (no minus sign).