# Integration - changing the limits

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: $$f= \dfrac{\sin(\sqrt{(x-\bar{x})^2+(y-\bar{y})^2})}{(x-\bar{x})^2+(y-\bar{y})^2}$$

Under the transformation $\bar{x}=x+u$ and $\bar{y}=y+v$ $\Rightarrow$ $f$ becomes $f(u,v)$ but the limits of integration with respect to $u$ and $v$ change.

Considering only the $x$ part: $$\int_{-x} ^{a-x}\int_0 ^a \sin(x)\sin(x+u)f(u,v) \, dx \, du$$ Integrating this with respect to $u$ analytically is difficult while with respect to $x$ is easier. But the $u$ limit have $x$ terms. Can I, by any way, make the limits of $u$ as $0$ to $a$ but yet carry a simpler integration with respect to $x$?

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What is the question? – Aryabhata Jan 25 '11 at 18:41
question: can I by anyway, make the limits of $u$ as $0$ to $a$ but yet carry a simpler integration with respect to $x$? (changing the limits of $x$ would cause no harm) – vijay Jan 26 '11 at 5:04