Integrate 1/√x(1+x) dx

Answer 1

Let's start by making a substitution:
u = 1 + x (so du/dx = 1 and dx = du)

Then the integral becomes:
∫ 1/√x(1+x) dx = ∫ 1/√u du

Using the power rule of integration for u^(n+1)/(n+1) with n=-1/2:
∫ 1/√u du = 2√u + C

Substituting back to the original variable x:
2√(1+x) + C = 2√(x+1) + C

Therefore, the indefinite integral of 1/√x(1+x) dx is 2√(x+1) + C.