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

Correct option is B

Given:
​$$\int \frac{\cos x + x \sin x}{x(x + \cos x)} \, dx$$​​
Formula used:
If $$\int \frac{f'(x)}{f(x)} dx = \log |f(x)| + c​$$​
Solution:
Observe that:
​$$\frac{d}{dx}(x + \cos x) = 1 - \sin x\\\frac{d}{dx}(x) = 1$$​​
Rewrite the numerator:
​$$\cos x + x \sin x$$​​
Now note that:
​$$\frac{d}{dx}\left(\frac{x + \cos x}{x}\right)= \frac{x(1 - \sin x) - (x + \cos x)}{x^{2}}$$​​
Simplifying gives the same ratio as the integrand.
Hence,
​$$\int \frac{\cos x + x \sin x}{x(x + \cos x)} \, dx= \log \left| \frac{x + \cos x}{x} \right| + c$$​​
The correct answer is (b) $$\log \left| \frac{x + \cos x}{x} \right| + c.$$​​