Integrating a Rational Function with a Cubic Denominator

[Ocis]

Determine the integral (x^2) / (2x^3 - 3)^2

This is probably easier than I expect but I have been attempting it using substitution...?
u = (2x^3 -3)
du/dx = 6x^2
dx = du / 6x^2

x^2 / (u^2) . du / 6x^2 - then I lose it because of the canceling out, and I'm unsure what happens to the x's...

I know the answer is -1/6 (2x^3 - 3) + C, but would appreciate the working I miss out. Or if I am attempting it using the wrong methods.

Many thanks,

Ocis

 

[Vagrant]

you get \int1/6u^2 du, so now integrate 1/u^2 with respect to u

 

[Ocis]

Thank you for the reply.

Ok so can I say that 1/u^2 = u^-2 ?
if so it would become 1/6 . (u^-1 / -1) = -1/6 (u) ? = -1/6 (2x^3 - 3) + C

Ok if that is right I think I understand a little more but just get confused with what to do with the x^2 as the numerator...?

 

[Vagrant]

dx=du/6x^{2}
when you substitute this in the expression the x^2 in the numerator cancels out, so there is in fact no x left in the expression.

 

[Ocis]

Ok I understand now, thank you very much.