Mathematics

Evaluate:
$$\int { \cfrac { \cos { 2x } +2\sin ^{ 2 }{ x }  }{ \sin ^{ 2 }{ x }  }  } dx$$


SOLUTION
Given $$\displaystyle\int{\dfrac{\cos{2x}+2{\sin}^{2}{x}}{{\sin}^{2}{x}}dx}$$

$$=\displaystyle\int{\dfrac{2{\cos}^{2}{x}-1+2{\sin}^{2}{x}}{{\sin}^{2}{x}}dx}$$    $$[\because \cos2x=2\cos^2-1]$$

$$=\displaystyle\int{\dfrac{2\left({\sin}^{2}{x}+{\cos}^{2}{x}\right)-1}{{\sin}^{2}{x}}dx}$$

$$=\displaystyle\int{\dfrac{2-1}{{\sin}^{2}{x}}dx}$$ since $$\left({\sin}^{2}{x}+{\cos}^{2}{x}=1\right)$$

$$=\displaystyle\int{\dfrac{1}{{\sin}^{2}{x}}dx}$$

$$=\displaystyle\int{{\csc}^{2}{x}dx}$$

$$=-\cot{x}+c$$ 

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Subjective Medium Published on 17th 09, 2020
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