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I have an integral $$\int_0 ^a\int_0 ^b\int_0 ^a\int_0 ^b \sin(x)\sin(\bar{x})\sin(y)\sin(\bar{y})f(x,\bar{x},y,\bar{y})~dx~dy~d\bar{x}~d\bar{y}$$ where $f= \dfrac{\sin\left(\sqrt{(x-\bar{x})^2+(y-\bar{y})^2}\right)}{(x-\bar{x})^2+(y-\bar{y})^2}$.

Under the transformation $$\bar{x}=x+u,~~~\text{and}~~~\bar{y}=y+v$$ the function $f$ becomes $f(u,v)$. This integral results in $$\int_0 ^a\int_0 ^b[(a-u)\cos(u)+\sin(u)][(b-v)\cos(v)+\sin(v)]f(u,v) \, du \, dv$$ Can someone help me in proving this?

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By proving, you mean solving the integral? – Torsten Hĕrculĕ Cärlemän Jul 23 '13 at 17:34
Ya, solving the integral. – vijay Jul 24 '13 at 5:43

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