Integration with change of variables $(x,y)\mapsto (x,xy)$?

Question

It is clear that $$\int_{[0,1]\times[0,1]}dxdy=1.$$ Let $z=xy$ then $(x,y)\mapsto (x,z)$ is injective from $[0,1]^2$ to $[0,1]^2$. The Jacobian is $J=\frac{1}{x}$. Then we have the integral $$\int_{[0,1]\times[0,1]}\frac{1}{x}dxdz=\infty.$$ What is going wrong?

Answer 1

$$\int_{x=0}^{x=1}\int_{y=0}^{y=1} dy\,dx=\int_{x=0}^{x=1}\int_{\frac zx=0}^{\frac zx=1}\frac1x\,dz\,dx=\int_{x=0}^{x=1}\int_{z=0}^{z=x}\frac1x\,dz\,dx=\int_{x=0}^{x=1}\frac xx\,dx.$$

Answer 2

$J(x,y)=(x,xy), Jac=\pmatrix{1&y\cr0&x}$ the Jacobian is $x$