Write $\int_{B_1(0)}\lambda^2({\rm d}x)\frac1{\sqrt{1-|x|^2}}h\left(\frac x{\sqrt{1-|x|^2}}\right)$ as an integral over $\mathbb R^2$

Question

Let $\lambda^2$ denote the Lebesgue measure on $\mathcal B(\mathbb R^2)$, $B_1(0)$ denote the unit ball around $0\in\mathbb R^2$, $$\varphi:B_1(0)\to\mathbb R^2\,,\;\;\;x\mapsto\frac x{\sqrt{1-|x|^2}}$$ and $h:\mathbb R^2\to[0,\infty)$ be Borel measurable. Can we write $$\int_{B_1(0)}\lambda^2({\rm d}x)\frac1{\sqrt{1-|x|^2}}(h\circ\varphi)(x)$$ as an integral over $\mathbb R^2$?