Find the value of $\int\int_{E} e^{x+y}dxdy$ where $E:$ Region lying between the two squares of side 2 and 4 centered at origin and sides parallel to the axes.
The required value will obviously be
$\int_{-2}^{2}\int_{-2}^{2}e^{x+y}dxdy - \int_{-1}^{1}\int_{-1}^{1}e^{x+y}dxdy$
(Subtracting the smaller square from the larger one to get the region between them)
$$=(e^{2}-e^{-2})^2 - (e^{1}-e^{-1})^2 = 4\cdot(\sinh(2)-\sinh(1))(\sinh(2)+\sinh(1))$$
The book lists the answer as $4\sinh(3)\sinh(1)$
Is there a way to simply further to obtain this result?
You're almost there:
$$ \begin{align} \dots &= \left(e^{2}-e^{-2}\right)^2 - \left(e-e^{-1}\right)^2\\ &= e^4-2+e^{-4}-e^2+2-e^{-2}\\ &= e^4-e^2-e^{-2}+e^{-4}\\ &= e^3\left(e-e^{-1}\right)-e^{-3}\left(e-e^{-1}\right)\\ &= \left(e^3-e^{-3}\right)\left(e-e^{-1}\right)\\ &= 4\sinh(3)\sinh(1) \end{align} $$
PS: to be honest, I figured it out backwards