Show that for all functions $f, g \in C^{2}(\mathbf{R})$ such that the functions $f, f^{\prime}, f^{\prime\prime}$ and $g$ have slow growth, we have
$$\int_{-\infty}^{+\infty} L(f)(x) g(x) \varphi(x) \mathrm{d}x = -\int_{-\infty}^{+\infty} f^{\prime}(x) g^{\prime}(x) \varphi(x) \mathrm{d}x$$
where $\forall x \in \mathbf{R}, \quad L(f)(x) = f^{\prime\prime}(x) - x f^{\prime}(x).$