I've somehow a really hard time showing that a distribution in $\mathcal{D'}(\mathbb{R^n})$ is regular.

The definition seems straight forward:

$ f \in \mathcal{D'}(\mathbb{R^n}) \text{ is regular if } \ \forall \varphi \in\mathcal{D}(\mathbb{R^n}) \ \exists g\in {L}_{loc}^{1}(\mathbb{R^n}) \ s.t. f(\varphi) = \int_{\mathbb{R}^n} g(x) \varphi(x) \ dx \\$ .

But I just don't see how one gets this function $g$.

For example one has given the function $f_n(x)= \frac{1}{n \pi} \frac{sin^2(x)}{x^2}$.

I mean I get the goal, it's now to find an integral such that $f_n(\varphi) = \int g_n(x) \varphi(x)$ dx. But I just don't know how and where to start.

For the longest time I thought it is the same as to show that $f_n$ is locally integrable, but I was told it's (completely) wrong.