Take a look at an online integrator, and I think you'll see this isn't going to be easy by hand.
The final value is 2.20335
If I had to change variables, I'd do
x = sinθ
dx = cosθ dθ
sqrt((1-x^2)(2-x)) dx = sqrt(cos^2θ*(2-sinθ) * cosV dθ
= cos^2θ*sqrt(2-sinθ) dθ
Not too up on elliptic integrals
Perform a change of variable and integrate the following elliptical integral
I = Square root [(1-x^2)*(2-x)]
limits
lower,-1 and
upper +1
could you show working
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