Double Integral Calculation with Variable Substitution

[physmatics]

Homework Statement

Calculate the double integral over D
$$\int\int x*ln(2x + y)/y^3 dx dy$$
D is the finite area in the xy-plane within the straight lines
$$2x + y = 1$$
$$2x + y = 3$$
$$x = y$$
$$x = 2y$$

Homework Equations

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The Attempt at a Solution

I thought it was obvious to make the variable substitution
$$u = x/y v = 2x + y$$
which gives us the boundaries
$$1 \leq u \leq 2 1 \leq v \leq 3$$
So far so good. Now, the problem is that I can't really substitute the whole integral. I thought this would solve itself by the Jacobian, but it turns out to be
$$y^2/2x + y$$
which I really don't need.
Should I really use this variable substitution? I don't know what else to do.

 

[tiny-tim]

hi physmatics!

have you tried just substituting v = 2x + y, and keeping y as the other variable?

 

[physmatics]

I wrote the Jacobian as a fraction instead, it turned out to work perfectly :)
Should have looked closer at it!