Evaluate the following integral : `int_0^(2pi)sqrt(1+sin(x/2))dx`

Explanation

`I=int_{0}^{2 pi} sqrt{1+sin (frac{{x}}{2})} d {x}`
`

sin ^{2}(frac{x}{2})+cos ^{2}(frac{x}{2})=1`
`

And sin (frac{x}{2})=2 sin (frac{x}{4}) sin (frac{x}{4})`


`I=int_{0}^{2 pi} sqrt{sin ^{2}(frac{x}{2})+cos ^{2}(frac{x}{2})+2 sin (frac{x}{4}) cos (frac{x}{4})} d x `
`
I=int_{0}^{2 pi} sqrt{.(sin (frac{x}{4})+cos (frac{x}{4}))^{2})} d x `
`
...