The ∫cos⁡x+xsin⁡xx(x+cos⁡x)dx=

∫cos⁡x+x+xsin⁡x−xx(x+cos⁡x)dx =∫cos⁡x+xx(x+cos⁡x)dx+∫x(sin⁡x−1)x(x+cos⁡x)dx=∫1xdx+∫sin⁡x−1x+cos⁡xdx=log⁡x−log⁡(x+cos⁡x)+c=log⁡xx+cos⁡x+c

The ∫cos⁡x+xsin⁡xx(x+cos⁡x)dx=

∫cos⁡x+x+xsin⁡x−xx(x+cos⁡x)dx =∫cos⁡x+xx(x+cos⁡x)dx+∫x(sin⁡x−1)x(x+cos⁡x)dx=∫1xdx+∫sin⁡x−1x+cos⁡xdx=log⁡x−log⁡(x+cos⁡x)+c=log⁡xx+cos⁡x+c

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The $\int \frac{\cos x + xsin x}{x (x + \cos x)} dx =$

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

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

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

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

answer is C.

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Detailed Solution

$\begin{array}{l} \int \frac{\cos x + x + xsin x - x}{x (x + \cos x)} dx \\ = \int \frac{\cos x + x}{x (x + \cos x)} dx + \int \frac{x (\sin x - 1)}{x (x + \cos x)} dx \\ = \int \frac{1}{x} dx + \int \frac{\sin x - 1}{x + \cos x} dx \end{array}$
$\begin{array}{l} = \log x - \log (x + \cos x) + c \\ = \log (\frac{x}{x + \cos x}) + c \end{array}$