Mathematics

Evaluate the following integral$\int { \cfrac { 1+\cot { x } }{ x+\log { \sin { x } } } } dx\quad$

SOLUTION
Let
$t=x+\log{\sin{x}}\Rightarrow\,dt=\left(1+\dfrac{1}{\sin{x}}\cos{x}\right)dx=\left(1+\cot{x}\right)dx$

$\displaystyle\int{\dfrac{\left(1+\cot{x}\right)dx}{x+\log{\sin{x}}}}$

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

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

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

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

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