Let $U,V \subset \mathbb{R}^n$ be open, and let $f: U \to V$ be a continuously differentiable bijection with continuously differentiable inverse. If $g: V \to \mathbb{R}$ is measurable and integrable, then
\begin{align*}
\int_V g(y)\,d\mathcal{L}^n(y)
&= \int_U g(f(x))\, |\det Jf_x|\, d\mathcal{L}^n(x).
\end{align*}