Question

Each expression that follows is the result of a calculation that uses integration by parts. That is, each is an expression of the form $uv-\int vdu$for some functions u and v. Identify u, v, du, and dv, and determine the original integral u dv.

$-x^2\cos x+2\int x\cos xdx$

Short answer

$\begin{array}{rcl}u& =& x^2\\ v& =& -\cos x\\ du& =& 2x\\ dv& =& \sin x\\ \int udv& =& \int x^2\sin xdx\end{array}$

Step-by-step solution

Step 1. Given information

Integral is$-x^2\cos x+2\int x\cos xdx$

Step 2. Explanation

Take $u=x^2,v=-\cos x$

$\begin{array}{rcl}du& =& 2xdx\\ dv& =& \sin xdx\\ \int udv& =& \int x^2\sin xdx\end{array}$