use the change of variables x = (u-v)/SQRT(2), y = (u+v)/SQRT(2) to evaluate

[integral from 0 to 1] [integral from 0 to 1] 1/(1-xy) * dxdy

i am given the hint to use the identity (1 - sint)/cost = cost/(1+sint) = tan(pi/4 -t/2)

so i need to prove the identities (have done so...no problem with that so i wont post it), transform the integral into terms of u and v, and then integrate!

the domain in xy is a square in the first quadrant, vertices (0,0) (0,1) (1,0) and (1,1)

let L_{1}be the bottom edge of the square

y = 0, 0 <= x <=1

i.e. v = -u, 0 <= u <= 1/SQRT(2)

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let L_{2}be the right edge of the square

0 <= y <= 1, x = 1

i.e. v = u - SQRT(2), 1/SQRT(2) <= u <= SQRT(2)

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let L_{3}be the top edge of the square

y = 1, 0 <= x <=1

i.e. v = SQRT(2) - u, 1/SQRT(2) <= u <= SQRT(2)

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let L_{4}be the left edge of the square

0 <= y <= 1, x = 0

i.e. v = u, 0 <= u <= 1/SQRT(2)

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so when all is said and done, the transformation from the domain D relating to xy to the domain T relating to uv is a square in the uv plane with vertices (0,0) (1/SQRT(2), 1/SQRT(2)) ((SQRT(2), 0) and (1/SQRT(2), -1/SQRT(2))

also the Jacobian is 1

please if you can double check this, as I'm not positive I got it right

now, from here I can't figure out how to evaluate the integral! Please help if you can, thanks!!