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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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