Mathematics

Evaluate the following integral
$$\int { \cfrac { \sec { x } \, cosec { x }  }{ \log { \left( \tan { x }  \right)  }  }  } dx$$


SOLUTION
$$I=\displaystyle\int{\dfrac{\sec{x} \, cosec{x}dx}{\log{\left(\tan{x}\right)}}}$$

Let

$$t=\log{\left(\tan{x}\right)}\Rightarrow\,dt=\dfrac{{\sec}^{2}{x}}{\tan{x}}dx$$

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

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

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

$$=\log{t}+c$$

$$=\log{\log{\left(\tan{x}\right)}}+c$$ where $$t=\log{\left(\tan{x}\right)}$$
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Subjective Medium Published on 17th 09, 2020
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