By means of a suitable subsituition, or otherwise, evaluate?

Question

/ 3
|          x/(2x - 1)^3 dx
/ 1

I need help with the suitable subsituition. I can do the otherwise method. I'm just interested in the suitable subsituition. Thanks in advance.

Answer 1

The only substitution that seems even marginally worthwhile is u = 2x - 1, du = 2dx to give
∫(1 to 5) u^(-3).(u+1)/2 (1/2) du
= ∫(1 to 5) (u^(-2) + u^(-3)) / 4 du
= [(-u^(-1) - u^(-2)/2) / 4] [1 to 5]
= (-1/5 - 1/50) / 4 - (-1 - 1/2) / 4
= -8/25.