Mathematics

Evaluate the given integral.
$$\displaystyle\int { \cfrac { 1 }{ \sin ^{ 2 }{ x } \cos ^{ 2 }{ x }  }  } dx$$


SOLUTION
$$I=\displaystyle\int{\dfrac{dx}{{\sin}^{2}{x}{\cos}^{2}{x}}}$$

$$=\displaystyle\int{\dfrac{4dx}{{4\sin}^{2}{x}{\cos}^{2}{x}}}$$

$$=\displaystyle\int{\dfrac{4dx}{{(2\sin}{x}{\cos}{x})^2}}$$

$$=\displaystyle\int{\dfrac{4\,dx}{{\sin}^{2}{2x}}}$$

$$=4\displaystyle\int{{cosec}^{2}{2x}dx}$$

$$=\dfrac{4\cos{2x}}{2}+c$$   .......since $$\displaystyle\int{{cosec}^{2}{ax}dx}=\dfrac{1}{a}\cot{ax}+c$$

$$=2\cos{2x}+c$$
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Subjective Medium Published on 17th 09, 2020
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