Evaluate: x - x x x ( x+ x) dx - Vidyadip

Question

Evaluate:
$\int\frac{\sin \text{x} - \text{x}\cos \text{x}}{\text{x} ( \text{x}+ \sin \text{x})} \text{dx}$

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Answer

$\text{I} = \int\frac{\text{x}+ \sin \text{x - x }(1 + \cos \text{x})}{\text{x} (\text{x} + \sin \text{x})}\text{dx}$
$= \int\frac{1}{\text{x}} \text{dx} - \int\frac{1 + \cos \text{x}}{\text{x}+ \sin \text{x}} \text{dx put x} + \sin \text{x} = \text{t} $
$\Rightarrow ( 1 + \cos \text{x}) \text{dx = dt}$
$= \log|\text{x}| - \log|\text{x} + \sin \text{x}| + \text{c} $

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