# Double Integral Of A Sine Function Using Change of Variable

I have to calculate the integral $$\int_0^x \int_x^{2\pi} \left( \frac{ \sin(N(u-v)/2)}{\sin((u-v)/2)} \right)^2 du dv$$ in terms of $$x$$ and was given a hint to use change of variable substitution: $$a =u-v, b= u+v$$.

However, I'm unsure what the new bounds would be after such a substitution. Currently, we are integrating over a rectangle where $$v$$ spans from $$0$$ to $$x$$ and $$u$$ from $$x$$ to $$2\pi$$. But, I can't figure out how to express this same rectangle in terms of $$u-v$$ and $$u+v$$.

I know that $$u-v$$ takes on values from $$0$$ to $$2\pi$$ and $$u+v$$ goes from $$x$$ to $$2 \pi + x$$. However, we need more conditions than that because the resulting rectangle that forms from these bounds is not equivalent to what we had before.

Any help would be greatly appreciated!