Difficult Integral Problem

Question

How can I find the definite integral of ((x^2-1)/sqrt(2x-1) )dx?

i tried using subsitution and solving for x but i cant come down to a definite answer.

Answer 1

let u = 2x - 1
du = 2 dx

x = ½ (u + 1)

½ Int[ (¼ (u + 1)²  - 1) / u^(½) ]
(1/8) Int[ (u² + 2u + 1 - 4) / u^(½) ]
(1/8) Int[ u^(3/2) + 2u^(1/2) - 3u^(-1/2) ]

(1/8) [ (2/5)u^(5/2) + 2(2/3)u^(3/2) - 3(2)u^(1/2) ]

(1/8) [ (2/5)(2x - 1)^(5/2) + (4/3)(2x - 1)^(3/2) - 6(2x - 1)^(1/2) ]