Question

Integrate the function.
$\int xsin3xdx.$

Answer 1

Let $I=\int xsin3xdx$
On taking x as first function and sin 3x as second function and integrating by parts, we get
$I=x\int sin3xdx-\int \left[\frac{d}{dx}\right(x)\int sin3xdx]dx$
$=\frac{-xcos3x}{3}+\int \frac{cos3x}{3}dx(\because \int sinaxdx=\frac{-cosax}{a})\Rightarrow I=\frac{-xcos3x}{3}+\frac{1}{9}sin3x+C$