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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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