Question

Evaluate $\int x^2\sin xdx$

• A. $-x^2\cos x+2(x\sin x+\cos x)$
• B. $-x^2\cos x-2(x\sin x+\cos x)$
• C. $x^2\cos x+(2x\sin x+\cos x)$
• D. $x^2\cos x-(2x\sin x-\cos x)$

Answer 1

The correct option is A $-x^2\cos x+2(x\sin x+\cos x)$

Given, $\int x^2\sin xdx$
Using Integration by parts method we have,
$\int x^2\sin xdx=x^2\int \sin xdx-\int \frac{d{x}^2}{dx}(\int \sin xdx)dx$
$\Rightarrow -x^2\cos x+2\int x\cos xdx$
$\Rightarrow -x^2\cos x+2[x\int \cos xdx-\int \frac{dx}{dx}\int \left(\cos xdx\right)dx]$
$\Rightarrow -x^2\cos x+2(x\sin x-\int \sin xdx)$
$\Rightarrow -x^2\cos x+2(x\sin x+\cos x)$