Question

Integrate cos^{​​​​4}​​​2x

Answer 1

I = ∫ cos⁴ 2x dx = ∫ (cos² 2x)² dx

Use the double angle formulae that cos 2A = 2 cos² A - 1 with A = 2x
2 * I = ∫ (1 + cos 4x)² dx
2 * I = ∫ 1 + 2 cos 4x + cos² 4x dx

Use the double angle formulae that cos 2A = 2 cos² A - 1 with A = 4x
4 * I = ∫ 2 + 4 cos 4x + [1 + cos 8x] dx
4 * I = ∫ 3 + 4 cos 4x + cos 8x dx

32 * I = ∫ 24 + 32 cos 4x + 8 cos 8x dx

32 * I = 24x + 8 sin 4x + sin 8x + constant

I = (3/4) x + (1/4) sin 4x + (1/32)sin 8x + constant