Question

Solve:
$\int \sin x\left(2+5x\right)dx$

Answer 1

$\int \sin x(2+5x)dx=\int (2\sin x+5x\sin x)dx$

$=\int 2\sin xdx+\int 5x\sin xdx$

$=-2\cos x+5\left[x\int \sin xdx-\int \left\{\frac{d}{dx}\right(x\left)\int \sin xdx\right\}dx\right]$

$=-2\cos x+5\left[x\right(-\cos x)-\int (-\cos x)dx]$

$=-2\cos x+5[-x\cos x+\int \cos xdx]$

$=-2\cos x-5x\cos x+5\sin x+c$

$\int \sin x(2+5x)dx=5\sin x-2\cos x-5x\cos x+c$.