Binomial series with integral

[sambellamy]

The problem I'm working on asks to "expand the integral as a binomial series". I have:

ʃ₀^π/2 (1-k²sin²(x))^-1/2 dx

It also gives k=sin(1/2 Θ₀)

Should I substitute this first? should I take the integral? should I try to simplify something? Is it ok to have sin(1/2 Θ₀)sin²(x) for x in the binomial series? Help! Thanks!

 

[Ishuda]

It is ok to have k² sin²(x) for the x in the binomial series but to avoid confusion, lets call it u. That is
\(\displaystyle (1 - u)^{-\frac{1}{2}}\) ~ \(\displaystyle 1 + \frac{1}{2}u = 1 + \frac{1}{2} k^2 sin^2(x)\)
and so
\(\displaystyle \int [1 - k^2 sin^2(x)]^{-\frac{1}{2}}\) ~ \(\displaystyle \int [ 1 + \frac{1}{2} k^2 sin^2(x) ]\)