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

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Solution

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

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