Mathematics

Evaluate the following integral
$$\int { \cfrac { cosec ^{ 2 }{ x }  }{ 1+\cot { x }  }  } dx\quad \quad $$


SOLUTION
Let 
$$t=1+\cot{x}\Rightarrow\,dt=-{\csc}^{2}{x}dx$$

$$\displaystyle\int{\dfrac{{cosec}^{2}{x}\,dx}{1+\cot{x}}}$$

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

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

$$=-\log{\left|1+\cot{x}\right|}+c$$ where $$t=1+\cot{x}$$
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Subjective Medium Published on 17th 09, 2020
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