Q 20. - Calculus

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 $uv - \int v du$ for some functions u and v. Identify u, v, du, and dv, and determine the original integral  u dv.

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

Step-by-Step Solution

Answer

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

Step 1. Given information

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

Step 2. Explanation

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

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