Science:Math Exam Resources/Courses/MATH103/April 2016/Question 02 (b)/Solution 1

Recall the integration by parts formula

$\int u\,dv=uv-\int v\,du.$

For this integral, we take $u=x$ and $dv=\cos(x)\,dx$, so $du=dx$ and $v=\sin(x)$. Applying the formula, we obtain

${\begin{aligned}\int x\cos(x)\,dx&=x\sin(x)-\int \sin(x)\,dx\\&=\color {blue}x\sin(x)+\cos(x)+C.\end{aligned}}$