Science:Math Exam Resources/Courses/MATH105/April 2018/Question 01 (i)/Solution 1

We will integrate by parts, with ${\textstyle u(x)=x}$ and ${\textstyle v'(x)=\sin x}$. Then, we have $u'(x)=1$ and $v(x)=-\cos x$, and the integration by parts formula gives

$$\int x\sin x\,dx=-x\cos x+\int \cos x\,dx=-x\cos x+\sin x+C,$$

where ${\textstyle C}$ is a constant.

Answer: $\color {blue}\sin x-x\cos x+C.$