Help with integration

Question

The first part of the question was proving the identity sin2x/1+tan²x is equal to 2sinxcos³x . I managed to do this by using Pythagorean identities.

The next part says "hence integrate 4sin4x/ 1+tan²2x

But I'm not sure how to use the first part of the question in the second part of the question

Answer 1

We can use the first part to substitute in to make

4sin4x/(1 + tan²2x) = 4(2sin2xcos³2x) = 8sin2xcos³2x

The leap: Consider the derivative of cos2x (using the chain rule):

d/dx(cos2x) = 2sin2x

So if we use a substitute function

w = cos2x, then

dw = 2sin2x

Now sub in w and dw into the expression above:

8sin2xcos³2x = 4w³dw

The integral of that is simply

w⁴ or cos⁴2x