Mathematics

The integral $$\displaystyle  \int _{ \dfrac { \pi  }{ 4 }  }^{ \dfrac { 3\pi  }{ 4 }  }{ \dfrac { dx }{ 1+\cos { x }  }  }$$


ANSWER

$$2$$


SOLUTION
Multiply Numerator & denominator by $$ (1 - cos x) $$

$$ \displaystyle \int \dfrac{1 - \cos x}{(1 + \cos x) (1 - \cos x)}dx = \int \dfrac{1 -\cos x}{1 -\cos x}dx $$ 

$$ \displaystyle = \int \dfrac{1 - \cos x}{\sin ^2 x}dx  = \int \dfrac{1}{\sin^2 x}dx  - \int \dfrac{\cos x}{\sin^2 x}dx $$

$$ \displaystyle = - \cot x - \int \dfrac{1}{y^2}dy \, (y = \sin x , dy = \cos x \, dx) $$

$$ \displaystyle = -\cot x + \dfrac{1}{y} = \left[-\cot x + \dfrac{1}{\sin x} + c\right]_{\frac{\pi}{4}}^{^3\frac{\pi}{4}} $$

$$ \displaystyle = \left[-\frac{1}{\tan x} + \frac{1}{\sin x} \right]^{^3\frac{\pi}{4}} - \left[-\frac{1}{\tan x} + \frac{1}{\sin x}\right]^{^3\frac{\pi}{4}}  $$

$$ \displaystyle =  (1 + \sqrt 2) - (-1 + \sqrt 2) = 2 $$


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Single Correct Medium Published on 17th 09, 2020
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