​$$\int \frac{\cos x + x \sin x}{x(x + \cos x)} \, dx$$ is equal to:​​

Given:
​$$\int \frac{\cos x + x \sin x}{x(x + \cos x)} \, dx$$​​​
Solution:
​​$$\begin{aligned}&\int \frac{\cos x + x \sin x}{x(x + \cos x)} \, dx \\&=\int \frac{\cos x + x \sin x+x-x}{x(x + \cos x)} \, dx \\&= \int \frac{(x + \cos x) - x(1 - \sin x)}{x(x + \cos x)} \, dx \\&= \int \left( \frac{x + \cos x}{x(x + \cos x)} - \frac{x(1 - \sin x)}{x(x + \cos x)} \right) \, dx \\&= \int \frac{1}{x} \, dx - \int \frac{1 - \sin x}{x + \cos x} \, dx \\&= \log |x| - \log |x + \cos x| + C \\&= \log \left| \frac{x}{x + \cos x} \right| + C\end{aligned}$$

The correct answer is (b) $$\log \left| \frac{x}{x + \cos x} \right| + C.$$​​