Mathematics

# Write a value of $\displaystyle\int { \cfrac { \cos { x } }{ \sin { x } \log { \sin { x } } } } dx\quad \quad$

##### SOLUTION
Let $t=\sin{x}\Rightarrow\,dt=\cos{x}dx$

$\displaystyle\int{\dfrac{\cos{x}dx}{\sin{x}\log{\sin{x}}}}$

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

Let $u=\log{t}\Rightarrow\,du=\dfrac{1}{t}dt$

$=\displaystyle\int{\dfrac{du}{u}}$

$=\log{\left(u\right)}+c$     ..........where $c$ is the constant of integration.

$=\log{\left(\log{t}\right)}+c$     ............where $u=\log{t}$

$=\log{\left(\log{\sin{x}}\right)}+c$      ..........where $t=\sin{x}$

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

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