Help with multi-variable calc

Use an appropriate change of variables to
evaluate the integral
I =double integral (xy) dxdy
when D is the region in the first quadrant
bounded by the lines
y = x , y = 5x ,
and the hyperbolas
xy = 3 , xy = 5 .

please help me with this.

Let u = y/x and v = xy.

So, y = x and y = 5x ==> u in [1, 5]
xy = 3 and xy = 5 ==> v in [3, 5].

Next, since ∂(u,v)/∂(x,y) =
|-y/x^2...1/x|
|...y...........x| = -2y/x = -2u,

we have |∂(x.y)/∂(u,v)| = |1/(-2u)| = 1/(2u).

Therefore, ∫∫D xy dA
= ∫(v = 3 to 5) ∫(u = 1 to 5) v * ((1/(2u)) du dv), via change of variables
= (1/2) ∫(v = 3 to 5) v dv * ∫(u = 1 to 5) du/u
= (1/2) * ((25 - 9)/2) * (ln 5 - ln 1)
= 4 ln 5.

I hope this helps!