How to calculate a double integral

Question

I’ve got the following integral $$\int\int _D \frac{dxdy}{x+y}$$

D is the region bounded by $x+y = 1, x+y = 4, y=0, x=0$ and I have to use the transformation $x = u-uv, y=uv$

Anyone know what domain to use and how to calculate the integral?

Answer 1

$x = u -uv$ , $y=uv$

$x_u = 1-v$ , $x_v = -u$

$y_u = v$ , $y_v = u$

$J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} 1-v & -u \\ v & u \end{vmatrix} = u -uv + uv = u$

So, $$dxdy = |J|dudv = u\ du\ dv$$

For regions,

$x+y = 1 \implies u =1$

$ x+y = 1 \implies u =4$

$y = 0 \implies uv =0 \implies u =0 $ or $v=0$

$x = 0 \implies u - uv=0 \implies u= 0$ or $v = 1$

Can you take it from here?