Question

Evaluate: $\int x\cdotp \sin 2x \ dx$

Answer 1

Solution:
To integrate the given function we must use integration by parts

Formula:

$\int U.V dx=U\int V dx-\int (\frac{dU}{dx}\int V dx)dx$

$\int\:x\:sin\:2xdx=x\int\:sin\:2x dx-\int (\frac{d\:x}{dx}\int\:sin\:2x dx)dx\\\\=\frac{-xcos\:2x}{2}+\int\frac{cos\:2x}{2} dx\\\\ =\frac{-xcos\:2x}{2}+\frac{sin\:2x}{4}$

So,

$\int\:x\:sin\:2xdx=\frac{-xcos\:2x}{2}+\frac{sin\:2x}{4}+C$