How do I integrate this function

The function is f ( x ) = (cos 2x) sqrt(1 + 4(sin 2x)^2)

How do I integrate it?

Start out by substituting:
u = sin(2x) ==> du = 2cos(2x) dx.

This gives:
∫ cos(2x)√[1 + 4sin^2(2x)] dx
= 1/2 ∫ √[1 + 4sin^2(2x)] [2cos(2x) dx], by re-writing
= 1/2 ∫ √(1 + 4u^2) du, by applying substitutions.

By integral tables, we can write:
1/2 ∫ √(1 + 4u^2) du
= 1/2 ∫ √[1 + (2u)^2] du
= (1/8)√[1 + (2u)^2] + (1/4)ln|2u + √[1 + (2u)^2]| + C
= (1/8)√(1 + 4u^2) + (1/4)ln|2u + √(1 + 4u^2)| + C.

Therefore, back-substituting yields:
∫ cos(2x)√[1 + 4sin^2(2x)] dx
= (1/8)√(1 + 4u^2) + (1/4)ln|2u + √(1 + 4u^2)| + C
= (1/8)√[1 + 4sin^2(2x)] + (1/4)ln|2sin(2x) + √[1 + 4sin^2(2x)]| + C.

I hope this helps!