^ /2 _ 0 ( 5 x + 3 x x + x )dx - Vidyadip

Question

$\int\limits^{\pi/2}_{0}\bigg(\frac{5 \sin \text{x} + 3 \cos \text{x}}{\sin \text{x} + \cos \text{x}}\bigg)\text{dx} $

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Answer

$\text{Let I} = \int\limits^{\pi/2}_{0}\bigg(\frac{5 \sin \text{x} + 3 \cos \text{x}}{\sin \text{x} + \cos \text{x}}\bigg)\text{dx}\dots\dots\dots\dots\text{(i)} $
$\Rightarrow\text{I}\ \ \ \ = \int\limits^{\pi/2}_{0}\frac{5 \cos \text{x} + 3 \sin \text{x}}{\cos \text{x} + \sin \text{x}}\text{dx}\dots\dots\dots\dots\text{(ii)} \Bigg(\therefore\int\limits^{\text{a}}_{0} \text{f(x)dx} = \int\limits^{\text{a}}_{0} \text{f(a - x)dx}\Bigg) $
Adding (i) and (ii)
$\text{2 I} = 8 \int\limits^{\pi/2}_{0} 1 . \text{dx} = 4\pi$
$\Rightarrow \text{I} = 2\pi$

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