Evaluate: ^ /4 _ 0 ( x + x 3 + 2x )dx - Vidyadip

Question

Evaluate:
$\int\limits^{\pi/4}_{0}\bigg(\frac{\sin \text{x} +\cos \text{x}}{3 + \sin 2\text{x}}\bigg)\text{dx}$

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Answer

$\text{I} = -\int\limits^\frac{\pi}{4}_{0} \frac{\sin \text{x} + \cos \text{x}}{(\sin \text{x} - \cos \text{x})^{2} - 2^{2}}\text{dx}$
$\text{Put} \sin \text{x} - \cos \text{x} = \text{t} \Rightarrow \text{t} = \text{-1 to } 0$
$(\cos \text{x} + \sin {\text{x)}} \text{dx = dt} $
$\text{I} = - \int\limits^{0}_{-1}\frac{\text{dt}}{\text{t}^{2} - 2^{2}}$
$= -\frac{1}{4}\log\bigg|\frac{\text{t} - 2}{\text{t} + 2}\bigg|\bigg]^{0}_{-1}$
$= -\frac{1}{4}\big\{0 - \log 3\big\}$
$= \frac{1}{4}\log3$

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