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Use the definition, there is a limit, write explicitly the limit in sage.

For instance, $$\int_{-1}^1\frac{dx}x$$ makes no sense as it is, but we may want to compute instead $$\text{PV}\int_{-1}^1\frac{dx}x =\text{PV}\int_{-1}^1\frac{dx}x := \lim_{a\searrow0}\text{PV}\int_{[-1,-a]\cup[a,1]}\frac{dx}x \ .$$ This can be easily implemented:

var('a,x')
assume(a>0);
assume(a<1);
limit( integral(1/x, x, -1, -a) + integral(1/x, x, a, 1), a=0 )

We get zero.