In some sense, that's what the theory of tempered distributions provides.
If $f$ is integrable, we have
$$\hat{f}(0) = \int_{\bf R} f(x) \, dx$$
If $f$ is a tempered distribution, the expression $\hat{f}$ still makes senses as a distribution and depends in a nice way on $f$. We may then quotient ${\mathscr S}'$ by the set of continuous functions that are zero at zero. This quotient contains the set ${\bf R}$ of standard values of integrals of integrable functions and gives a new target space for this extended integral.
So for example we get
$$\int_{\bf R} \sin(x)\, dx = i\pi(\delta_1 -\delta_{-1})$$
$$\int_{\bf R} x^k\, dx = 2\pi i^k \delta_0^{(k)}$$
and so on.
EDIT: Thinking about it, maybe you are refering to integration as defined in nonstandard analysis? See the following link.
https://math.stackexchange.com/questions/464565/definition-of-the-integral-in-non-standard-calculus