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

log ∣ ∣ ∣ \frac{x}{x + cos x} ∣ ∣ ∣ + c

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log ∣ ∣ ∣ \frac{x + cos x}{x} ∣ ∣ ∣ + c

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log ∣ ∣ ∣ \frac{1}{x + cos x} ∣ ∣ ∣ + c

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log | x + cos x | + c

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Solution

The correct option is B $log ∣ ∣ ∣ \frac{x}{x + cos x} ∣ ∣ ∣ + c$
$\int \frac{cos x + x sin x}{x (x + cos x)} d x = \int \frac{(x + cos x) - x (1 - sin x)}{x (x + cos x)} d x$

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

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

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