Let $X$ be a random variable. I am trying to prove a special case of LOTUS (assuming $g$ is increasing and differentiable) using transformation of PDF. I have in my proof arrived at the following integral:
$$\mathbb{E}[g(X)] = ... = \int_\mathbb{R} g(x) \cdot\bigg\vert \frac{1}{g'(g^{-1}(x))} \bigg\vert \cdot f_X(g^{-1}(x)) \ dx$$
where $f_X$ denotes the PDF of $X$. I want to make the substitution $ u = g^{-1}(x) $ hoping some of it will cancel out. However I am a bit unsure how to change the variables in the integral itself. My guess is that it looks like this:
$$ \int_\mathbb{R} u^{-1} \cdot\bigg\vert \frac{1}{g'(u)} \bigg\vert \cdot f_X(u)\cdot u' \ du$$
However this is not quite what I want. Also can someone tell me if it is correct or not? And why? That would really be helpful!
PS. I am aware of how to compute an integral using substitution but since we are never actually calculating the integral itself I don't believe it's the same.