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!



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