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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