integration - sqrt(5)/(1+ sin x)^2

Question

1) differentiate -cos x / (1 + sin x) with respect to x.

2) hence find the ∫√5/(1 + sin x) ^2 dx

My answer to part 1 is 1/(1 + sin x). How does part 1 used in part 2?

Answer 1

d/dx [-cosx / (1+sinx)] = [(1+sinx)(sinx) + (cosx)(cosx)] / (1 + sinx)²

= [1 + sin²x + cos²x] / (1 + sinx)² = 2 / (1 + sinx)²

Therefore, 2 / (1 + sinx)² is an antiderivative of -cosx / (1 + sinx)

So, √5∫[1 / (1 + sinx)²]dx = (√5/2)∫[2/(1 + sinx)²]dx

= (√5/2)[-cosx / (1 + sinx)] + C