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$\int \frac{\cos x + x \sin x}{x (x + \cos x)} d x =$

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$\log | x (x + \cos x) | + c$

$\log | x (\cos x + x \sin x) | + c$

$\log |\frac{x}{x + \cos x}| + c$

$\log |\frac{x + \cos x}{x}| + c$

answer is C.

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Detailed Solution

$\int \frac{\cos x + x \sin x}{x (x + \cos x)} d x$

$= \int \frac{(x + \cos x) - x (1 - \sin x)}{x (x + \cos x)} d x$

$= \int (\frac{1}{x} - \frac{1 - \sin x}{x + \cos x}) d x$

$= \log | x | - \log | x + \cos x | + c$

$= \log |\frac{x}{x + \cos x}| + c$

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