Question

help with an indefinite integral?

cos2x/((sin^2)(cos^2))

Answer 1

The integral equals `-2csc(2x) + C`

Explanation:

We have:

`I = int cos(2x)/(sin^2xcos^2x)dx`

Recall that `sin(2x)= 2sinxcosx`, therefore `sin^2(2x) = 4sin^2xcos^2x` and so `sin^2xcos^2x =1/4sin^2(2x)`.

`I = 4 int cos(2x)/(sin^2(2x))dx`

`I = 4int cot(2x)csc(2x)dx`

We let `u = 2x`. Then `du = 2dx` and `dx = (du)/2`.

`I = 4int cotucscu * (du)/2`

`I = 2int cotucscudu`

This is a known integral.

`I = -2cscu + C`

`I = -2csc(2x) + C`

Hopefully this helps!