Verified answer

integration by sub. u= 4-3 sin2ß.......(1) then du/dß=-6cosß, i.e

dß=du/(-6cosß)......(2)

substitue (1),(2) into the integrand you will get

∫(4-3 sin2ß)^4 cos 2 ßdß= ∫(u)^4* cos 2 ß du/(-6 cos 2 ß) =-1/6 u^5/5+c

=-1/30(4-3 sin2ß)^5+C

I = ∫ ( 4 - 3 sin 2β )^4 ( cos 2β ) dβ

Let u = 4 - 3 sin 2β

du = - 6 cos 2β

- du / 6 = cos 2β

I = ( - 1/6 ) ∫ ( u )^4 du

I = (- 1/30 ) u^5 + C

I = (- 1/30 ) (4 - 3 sin 2β)^5 + C

let u = 4-3 sin2ß the rest will fall out

let u = 4-3sin2ß --->du = -6cos2ß dß

answer = -(1/30)(4-3sin2ß)^5 + c