Definite integral calculus prove

Question

prove ,

∫₀^π/2(cos2x)² cosx dx=√2 ∫₀^π/4(sin2x)² cosx dx_

Answer 1

Because cos²2xcosx = 1/2(1 + cos4x)cosx = 1/2cosx + 1/4cos5x + 1/4cos3x;

So, ∫₀^π/2(cos2x)² cosx dx = (1/2sinx + 1/20sin5x + 1/12sin3x)₀^π/2 = 1/2 + 1/20 - 1/12 = 7/15.

Because sin²2xcosx = 1/2(1 - cos4x)cosx = 1/2cosx - 1/4cos5x - 1/4cos3x.

So,√2 ∫₀^π/4(sin2x)² cosx dx = √2(1/2sinx - 1/20sin5x - 1/12sin3x)₀^π/4 = √2/2(√2/2 -1/10·(- √2/2) - 1/6√2/2) =

1/2(1 + 1/10 - 1/6) = 7/15.

So we prove equality of these integrals.