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I do not know much about the notion of Cauchy principal value integral, but you can reproduce the two examples given on Wikipedia https://en.wikipedia.org/wiki/Cauchy_principal_value#Examples :

sage: a = SR.var('a')
sage: assume(0<2*a<1)
sage: f = integral(1/x, x, -1, -2*a) + integral(1/x, x, a, 1)
sage: f.limit(a=0)
log(2)

and

sage: a = SR.var('a')
sage: assume(a>0)
sage: f = integral(2*x/(x^2-1), x, -2*a, a)
sage: f.limit(a=oo)
-2*log(2)