Question

Make a substitution to express the integrand as a rational function and then evaluate the integral. (Use C for the constant of integration.)
1/x{(x-1)^1/2} dx

Answer 1

let
u^2 = x-1
x = 1+u^2
2u du = dx

Now your integrand is

1/(1+u^2) * u * 2u du
= 2u^2/(1+u^2) du
= 2(1 - 1/(1+u^2)) du

That integrates to

2(u - arctan(u))+C
= 2(√(x-1) - arctan√(x-1))+C