Mathematics

# Evaluate the following integral$\int { \cfrac { cosec { x } }{ \log { \tan { \cfrac { x }{ 2 } } } } } dx$

##### SOLUTION
Let
$t=\log{\tan{\dfrac{x}{2}}}\Rightarrow\,dt=\dfrac{1}{\tan{\dfrac{x}{2}}}{\sec}^{2}{\dfrac{x}{2}}\times\dfrac{1}{2}dx$

$\Rightarrow\,dt=\dfrac{1}{\tan{\dfrac{x}{2}}}{\sec}^{2}{\dfrac{x}{2}}\times\dfrac{1}{2}dx$

$\Rightarrow\,dt=\dfrac{1}{2}\dfrac{1}{{\cos}^{2}{\dfrac{x}{2}}}\times\cot{\dfrac{x}{2}}dx$

$\Rightarrow\,dt=\dfrac{1}{2}\dfrac{1}{{\cos}^{2}{\dfrac{x}{2}}}\times\dfrac{\cos{\dfrac{x}{2}}}{\sin{\dfrac{x}{2}}}dx$

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

$\Rightarrow\,dt=\dfrac{1}{\sin{x}}dx$

$\Rightarrow\,dt=cosec{x} \, dx$

Putting it in the integration we get,

$\displaystyle\int{\dfrac{\csc{x}}{\log{\tan{\dfrac{x}{2}}}}dx}$

$=\displaystyle\int{\dfrac{dt}{t}}$

$=\log{\left|t\right|}+c$

$=\log{\left|\log{\tan{\dfrac{x}{2}}}\right|}+c$ where $t=\log{\tan{\dfrac{x}{2}}}$

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