coordinate transformation double integral

Hey everyone, I am stuck on this question, I just can't get anymore done. Any help would be great.

Evaluate $\displaystyle \int_1^2 \int_{1/y}^y \ \sqrt{\frac{y}{x}} \cdot e^{\sqrt{xy}} \ dx \ dy$ by making the coordinate transformation $\displaystyle x=\frac{u}{v}$ and $\displaystyle y=uv$.

My working so far:

$\displaystyle xy=u^2$

$\displaystyle \frac{y}{x}=v^2$

Jacobian $\displaystyle \ = \frac{2u}{v}$

So:

$\displaystyle \int_1^2 \int_{1/uv}^{uv} \ \sqrt{v^2} \cdot e^{\sqrt{u^2}} \cdot \frac{2u}{v} \ du \ dv$

But I don't get how to go from here, with the limits of one of the integrals having terminals with u and v in it.

If anyone can help that would be fantastic!

Thanks