∫(sin x-xcos x)/(x(x+sin x))dx= (where C is an arbitrary constant)

The correct answer is Option (3) → $\log\left|\frac{x}{x+\sin x}\right|+C$

Given integrand:

$\displaystyle \frac{\sin x - x\cos x}{x(x+\sin x)}$

Rewrite as

$\displaystyle \frac{\sin x}{x(x+\sin x)}-\frac{\cos x}{x+\sin x}$

Note

$\displaystyle \frac{\sin x}{x(x+\sin x)}=\frac{1}{x}-\frac{1}{x+\sin x}$

So integrand $=\frac{1}{x}-\frac{1+\cos x}{x+\sin x}=\frac{d}{dx}\!\left[\ln\!\left(\frac{x}{x+\sin x}\right)\right]$

Therefore

$\displaystyle \int\frac{\sin x - x\cos x}{x(x+\sin x)}\,dx=\ln\left|\frac{x}{x+\sin x}\right|+C$