I'm trying to solve the following seemingly simple integral – so far, without success:
$$\int_{a}^{b}\int_{a}^{y}\frac{\cos(x-y)}{xy}\mathop{\mathrm{d}x}\mathop{\mathrm{d}y}$$
For some $0<a<b$. I've tried to change the order of integration, and integration by parts, but so far, without success. Solutions (also using $\mathop{\mathrm{Si}}$ & $\mathop{\mathrm{Ci}}$ functions) would be very appreciated.
Thanks!
Best Answer
Note that $$\begin{align}\int_{a}^{b}\int_{a}^{y}\frac{\cos(x-y)}{xy}\mathop{\mathrm{d}x}\mathop{\mathrm{d}y}&=\frac{1}{2}\int_{a}^{b}\int_{a}^{b}\frac{\cos(x-y)}{xy}\mathop{\mathrm{d}x}\mathop{\mathrm{d}y}\\ &= \frac{1}{2}\int_{a}^{b}\int_{a}^{b}\frac{\cos(x)\cos(y)+\sin(x)\sin(y)}{xy}\mathop{\mathrm{d}x}\mathop{\mathrm{d}y}\\ &=\frac{1}{2}\left(\int_{a}^{b}\frac{\cos(x)}{x}\right)^2 +\frac{1}{2}\left(\int_{a}^{b}\frac{\sin(x)}{x}\right)^2. \end{align}$$ then take a look to functions $\mathrm Si$ and $\mathrm Ci$.