this is an improper integral, so we have to use limits when evaluating it. for the anti derivative, we use substitution.

int{0-->infinity} [1/sqrt(x + 1)]dx

= int{0-->infinity} [(x + 1)^(-1/2)]dx

let u = x + 1

=> du = dx

so our integral becomes:

int{u^(-1/2)}du

= 2u^(1/2) + C

= 2(x + 1)^(1/2) + C

now to evaluate between 0 and infinity, we proceed like this

int{0-->infinity} [(x + 1)^(-1/2)]dx = lim{N-->infinity} int{0-->N} [(x + 1)^(-1/2)]dx

= lim{N-->infinity} 2(x + 1)^(1/2) evaluate between N and 0

= lim{N-->infinity} 2(N + 1)^(1/2) - 2(0 + 1)^(1/2)

= lim{N-->infinity} 2(N + 1)^(1/2) - 2

= infinity