Double integrals over the region [0,1] x [0,1].
The integrand is ** cos(2pi x)*cos(2pi
y)*[log((x-y)^2)-log(1+(x-y)^2)].** The latex code of
this function is given as follows: ($\int_0^1\int_0^1\cos 2\pi x\cos 2\pi
y\left(\log (x-y)^2-\log \left( 1+\left( x-y\right) ^2\right) \right) dxdy$)
Maple and Mathematica can't handle this function since the singular
points are located along x=y.

Double integrals over the region [-1,1] x [-1,1]. The integrand is
** f(x,y)=xy*(x^2+y^2)^(-2) if x^2 + y^2 > 0 and f(x,y)=0 if x^2 + y^2
=0. ** Both Maple and Mathematica give answers 0, which are wrong. Since
the double integral integral does not exist even if the repeated integrals
is 0. This example shows that the existence of a double integral is
essential for one to estimate its value since the existence and equality
of the repeated intgrals of a function (such as in this case) need not
imply that the double integral of the function exists.