Triangle integral ∫∫dxdyf(x*y) how to reduce to one dimension?

[VytautasD]

I meet with a triangle integral where x+y≤1, and function is dependant only on x*y. I am wondering if there any possibility to relate ∫∫dxdyf(x*y)=∫d(x*y)f(x*y)g(x*y) or something similar? Or maybe there are some assumptions needed to relate like this?

 

[LCKurtz]

I meet with a triangle integral where x+y≤1, and function is dependant only on x*y. I am wondering if there any possibility to relate ∫∫dxdyf(x*y)=∫d(x*y)f(x*y)g(x*y) or something similar? Or maybe there are some assumptions needed to relate like this?

I'm not sure this is what you want. But in the integral $$
\int_0^1\int_0^{1-x^2}f(xy)\, dydx$$you could change the inner variable from $y$ to $w$ with the substitution $w=yx,\ dw = x dy$ giving$$
\int_0^1\frac 1 x \int_0^{x-x^2}f(w)\, dwdx$$