Mathematics

# Evaluate the following integral$\int { \sqrt { \cfrac { 1-\cos { x } }{ 1+\cos { x } } } }$

##### SOLUTION
$I=\displaystyle\int{\sqrt{\dfrac{1-\cos{x}}{1+\cos{x}}}dx}$

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

$I=\displaystyle\int{\sqrt{\dfrac{{\sin}^{2}{\dfrac{x}{2}}}{{\cos}^{2}{\dfrac{x}{2}}}}dx}$

$I=\displaystyle\int{\dfrac{\sin{\dfrac{x}{2}}}{\cos{\dfrac{x}{2}}}dx}$

Let $t=\cos{\dfrac{x}{2}}\Rightarrow\,dt=\dfrac{-1}{2}\sin{\dfrac{x}{2}}dx$

$\Rightarrow\,-2dt=\sin{\dfrac{x}{2}}dx$

$I=-2\displaystyle\int{\dfrac{dt}{t}}$

$\Rightarrow\,I=-2\log{\left|t\right|}+c$

$\therefore\,I=-2\log{\left|\cos{\dfrac{x}{2}}\right|}+c$ where $t=\cos{\dfrac{x}{2}}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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