Mathematics

Evaluate the following integrals:
$$\displaystyle \int { \cot ^{ n }{ x }\ cosec ^{ 2 }{ x }  } dx,n\neq -1$$


SOLUTION
$$\displaystyle\int{{\cot}^{n}{x}\,{cosec}^{2}{x}dx}$$

$$=\displaystyle\int{{\left(\cot{x}\right)}^{n}{cosec}^{2}{x}dx}$$

Let 

$$t=\cot{x}\Rightarrow\,dt=-{cosec}^{2}{x}dx$$

$$=-\displaystyle\int{{t}^{n}dt}$$

$$=-\dfrac{{t}^{n+1}}{n+1}+c$$

$$=-\dfrac{{\cot}^{n+1}{x}}{n+1}+c$$ where $$t=\cot{x}$$
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Subjective Medium Published on 17th 09, 2020
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