Make a u substitution and integrate from u(a) to u(b) . u(a)=0 u(b)=30 cos to the negative 3 times 2 theta times sin 2 theta d theta

Question

Please solve  this problem today for me before 7pm EST. Thank you.

Answer 1

This reads ∫cos⁻³(2θ)sin(2θ)dθ for [0,30].

Let u=cos(2θ), du=-2sin(2θ)dθ, so sin(2θ)dθ=-½du.

The integral becomes -½∫u⁻³du=-½[-u⁻²/2]=¼u⁻² for u in [cos(0),cos(60)]=[1,½], where 1 is the low limit and ½ is the high limit.

This comes to ¼(4-1)=¾.

The limits apply to θ, not to u, because cosine cannot be outside the range [-1,1].