For the sample question on the Cauchy distribution,
the probability of error is the integral of P(x)*P(error_x), where
P(error_x) is min{P(Ci|X)}.  

Now, P(Ci|X) = P(Ci)*P(X|Ci) / P(x)    but P(Ci) is 0.5
Since the two P(X|Ci) are symmetric, we can just integrate from their
"midpoint" to +infinity, and multiply the result by 2.  So we just assume
that a_2 > a_1   i.e. C2 is more likely in the whole domain of
integration.  Thus we just use P(C1|X) in the integral, and after
substituting, we get an integral of P(X|C1) taken from
(a_2 + a_1)/2  to +infinity.
This integral gives the result that was asked for.
See if you can write down all the steps on your own.  The real lesson here
is to learn how to set up the probability of error, not how to integrate.