integrate cos^2(2x)sin^3(2x) dx

To integrate this we need to use the chain rule, substituting cos2x = u Integral becomes: u²sin³2x dxChain rule: dy/dx = du/dx dy/du du/dx = -2sin2x --> dx = -1/2sin2x du Substituting into the equation u²sin³2x * -1/2sin2x du Simplifies to: -2u²sin²2x duWe know that cos²x + sin²x =1 Integral = -2u²(1 - cos²2x) du Substituting -> -2u² + 2u⁴ duIntegrating this: -2( 1/3u³ - 1/5u⁵) + c Substituting u back into the equation: cos⁵2x/10 - cos³2x/6 + c