A distribution is tempered if and only if it is the distributional derivative $D^nF$ of a continuous function $F$ which is $O(x^k)$ for some positive integer. Its primitive is then $D^{n-1}F$ and so also tempered. Hence the answer to your question is no. (Suitable reference: J. Sebastiao e Silva, Integrals and orders of growth of distributions, Lisbon, 1964).

A distribution is tempered if and only if it is the distributional derivative $D^nF$ of a continuous function $F$ which is $O(x^k)$ for some positive integer. Its primitive is then $D^{n-1}F$ and so also tempered. Hence the answer to your question is no. (Suitable reference: J. Sebastiao e Silva, Integrals and orders of growth of distributions, Lisbon, 1964).

Edit. I wrote up the one-dimensional case dor simplicity. If $n=0$, you have a continous function and you simply take its classical primitive which is clearly a tempered distribution.