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-
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