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
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