Tempered distribution and primitive integral

Question

$f$ is a Schwartz function on $\mathbb{R}$. Define $g(x)= \int_{-\infty}^{x} f(x)dx$. Show that $g(x)$ is a tempered distribution.

Any ideas? I have no idea how to do the problem

Answer 1

Hint: if a function and all its derivatives have at most polynomial growth, then it defines a tempered distribution.

Now check that $g(x)$ satisfies the above hypothesis.