What's the integral of cos^5(2x)?

The integral of cos^5(2x) is (1/2)cos^4(2x)sin(2x) + (2/3)cos^3(2x)sin^3(2x) + C.

To solve this integral, we can use the substitution u = sin(2x) and du/dx = 2cos(2x). Then, we can rewrite cos^5(2x) as cos^4(2x)cos(2x) and substitute cos(2x) with (1 - u^2)^(1/2):

∫cos^5(2x)dx = ∫cos^4(2x)cos(2x)dx
Let u = sin(2x), du/dx = 2cos(2x), dx = (1/2)du/cos(2x)
∫cos^5(2x)dx = (1/2)∫(1-u^2)^2du
= (1/2)∫(1-2u^2+u^4)du
= (1/2)(u - (2/3)u^3 + (1/5)u^5) + C
Substituting back u = sin(2x):
= (1/2)sin(2x) - (2/3)sin^3(2x) + (1/5)sin^5(2x) + C
= (1/2)cos^4(2x)sin(2x) + (2/3)cos^3(2x)sin^3(2x) + C.