Use the substitution u=cos(2x)to find ∫(cos(2x))^2 (sin(2x))^3dx

Step 1 differentiate substitution: du/dx = -2sin(2x)Step 2 rearrange for dx: dx=du/-2sin(2x)Step 3 substitute: integral= ∫u²sin³(2x).du/-2sin(2x)Step 4 get the integral in terms of u by cancelling: integral=-o.5∫u²sin²(2x)dunote the identity sin²(2x)+cos²(2x)=1integral=-0.5∫u²(1-cos²(2x))du =-0.5∫u²(1-u²)du =-0.5∫u²-u⁴duStep 5 integrate: integral= -0.5(1/3u³-1/5u⁵)+cintegral= -1/6u³+1/10u⁵+cStep 6 replace u with substitution: integral= -1/6cos³(2x)+1/10cos⁵(2x)+c-