Question

Solve $\int x^2.\sin 2xdx$

• A. $\frac{x^2\cos 2x}{2}+\frac{2x\sin 2x}{4}+\frac{2\cos 2x}{8}+c$
• B. $\frac{{-x}^2\cos 2x}{2}+\frac{2x\sin 2x}{4}+\frac{2\cos 2x}{8}+c$
• C. $\frac{-x^2\cos 2x}{2}-\frac{x\sin 2x}{2}+\frac{\sin 2x}{8}+c$
• D. $None$

Answer 1

Answer: (B)

$\frac{{-x}^2\cos 2x}{2}+\frac{2x\sin 2x}{4}+\frac{2\cos 2x}{8}+c$

$\int x^2.\sin 2xdx$

$=x^2\times \frac{-\cos 2x}{2}-2x\times \frac{-\cos 2x}{2}dx$

$=\frac{-x^2\cos 2x}{2}+\int x\cos 2xdx$

$=\frac{-x^2\cos 2x}{2}+x\times \frac{\sin 2x}{2}-\int 1\times \frac{\sin 2x}{2}dx$

$=\frac{-x^2\cos 2x}{2}+x\times \frac{\sin 2x}{2}-\frac{1}{2}\times \frac{-\cos 2x}{2}+c$

$=\frac{-x^2\cos 2x}{2}+\frac{x\sin 2x}{2}+\frac{\cos 2x}{4}+c$

$=\frac{-x^2\cos 2x}{2}+\frac{2x\sin 2x}{4}+\frac{2\cos 2x}{8}+c$