It can be done either by differential method or the substitution
∫ 1/√(x + 1) dx
= ∫ (x + 1)^(- 1/2) d(x + 1)
= (x + 1)^(1 - 1/2)/(1 - 1/2)
= 2√(x + 1) + C
or let u^2 = x + 1 then 2u du = dx
∫ 1/√(x + 1) dx
= ∫ 1/u * (2u du)
= 2∫ du
= 2u
= 2√(x + 1) + C