Linear transformation of variable under the integral sign. Easy change of variables question I realize this might be a basic question, but I need a sanity check.
Let $f(\vec{x})$ be a function that takes $n$-dimensional vectors and returns a real number. Suppose the goal is to compute
$$\int_{\mathbb{R}^n} f(\vec{x})\, d\vec{x},$$
but in fact it is easier to work with the transformed variable $T\vec{x}$, where $T$ is some $n\times n$ matrix. What exactly is the change of variables procedure for 
$$\int_{\mathbb{R}^n} f(T(\vec{x}))\, d\vec{x}?$$
Do you simply divide the second integral by $\det(T)$?

The other answer uses the same name for the transformed variable, which can be confusing for some. Let's call $\vec{y} = T \vec{x}$. Then
\begin{align*}
    \int_{\mathbb{R}^n} f(\vec{x}) d \vec{x} &= \int_{\mathbb{R}^n} f(T^{-1}\vec{y}) \frac{1}{|\det T | } d \vec{y} 
\end{align*}
and also
\begin{align*}
    \int_{\mathbb{R}^n} f(T\vec{x}) d \vec{x} &= \int_{\mathbb{R}^n} f(\vec{y}) \frac{1}{|\det T|} d \vec{y}
\end{align*}
