Multivariable change of variable in integral

Let $f:\mathbb R^n\rightarrow [0,T]$ be surjective and smooth enough (you can assume more assuptions if you want). I'd like to calculate:

$$\int _{\mathbb R^n} g(f(x))dx$$ and $$\int _{\mathbb R^n} f(x)g(f(x))dx$$

where $g:[0,T]\rightarrow [0,T]$ smooth enough and increasing, by using change of variable and transform the integral into univariate one over $[0,T]$. How can this be done? I guess it might involve some gradients/determinants.

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