Edit: This questions has been sufficiently answered
I have a function which is too complicated to integrate using Riemann integration so I am trying to evaluate it with Lebesgue integration. I have seen how to integrate simple functions like $f(x)=x^2$ with Lebesgue integration but this involves using $f^{-1}(x)$.
The function I have does not have a closed form for its inverse. In fact the function I am considering is
$f(x)= \frac{x^{\frac{2}{3}}}{\sqrt{ax^{\frac{2}{3}}-1}}e^{(1-bx^{\frac{2}{3}}-cx^2)}$
Do I have any hopes of utilising Lebesgue integration to integrate this over all values of x which have real valued f(x)? The values of x where f(x) is real are $(a^{\frac{3}{2}},\infty)$