∫cosec 2 x(cos 2 x-sin 2 x) 2 dx Opening the square On multiplying (1-sin 2 x)(1-sin 2 x) equation reduces to =∫(cosec 2 x-2+sin 2 x-2cos 2 x+ sin 2 x)dx =∫(cosec 2 x-2+2sin 2 x-2cos 2 x)dx =∫(-2(cos 2 x-sin 2 x)+cosec 2 x-2)dx =∫(-2cos2x+cosec 2 x-2)dx On solving this we get our answer i.e =-sin2x-cotx-2x+c

Evaluate

∫cosec²x(cos²x-sin²x)²dx

Opening the square

On multiplying (1-sin²x)(1-sin²x) equation reduces to

=∫(cosec²x-2+sin²x-2cos²x+ sin²x)dx

=∫(cosec²x-2+2sin²x-2cos²x)dx

=∫(-2(cos²x-sin²x)+cosec²x-2)dx

=∫(-2cos2x+cosec²x-2)dx

On solving this we get our answer i.e

=-sin2x-cotx-2x+c