Question

cos²x/sin²2x,Integrate the given function w.r.t. x considering them well defined and integrable over proper domain.

Answer 1

In the attachment I have answered this problem.   The solution is simple and easy to understand.    See the attachment for detailed solution.

Answer 2

Dear Student,

Answer:$\int{\frac{ {cos}^{2}x }{ {sin}^{2}2x }}dx= - \frac{1}{4} cot \: x + c$

Solution:

$\int{\frac{ {cos}^{2}x }{ {sin}^{2}2x }}dx$

as we know that

${sin}2x = 2sinx \: cosx$

So,

$\int{\frac{ {cos}^{2}x }{ {(sin2x)}^{2} }}dx \\ \\ =\int{\frac{ {cos}^{2}x }{ {(2sinx \: cos \: x})^{2}}}dx \\ \\ = \int{\frac{ {cos}^{2} x}{4 {sin}^{2}x \: {cos}^{2} x}}dx \\ \\ \int{\frac{1}{4 {sin}^{2}x }} dx \\ \\ = \int{\frac{1}{4} {cosec}^{2} x} \: dx$

we know that

$\int{{cosec}^{2} x} dx = - cot \: x + c$

apply formula

$\int{\frac{1}{4} {cosec}^{2} x}dx = - \frac{1}{4} cot \: x + c$

Hope it helps you