calculus

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

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  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

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