Context and Assumptions : we consider the case of a very standard exercice in Schwartz functions theory: Let's suppose that $\phi$ is a $S(\mathbb{R})$ function, and let's suppose additionally that $\displaystyle\int_{-\infty}^\infty \phi(t)\ \mathrm dt=0$.
In a first time, I demonstrate that the function $\displaystyle \psi\colon\mathbb{R}\ni x\longmapsto \int_{-\infty}^x \phi(t)\ \mathrm dt\in \mathbb{R},$ defines a $S(\mathbb{R})$ function.
Then, I am asked to estime $N_p$ norm of $\psi$ (also called $N_p$ semi-norm) in function of $N_p(\phi)$.
For the reminder, it is defined by :
$$ N_p(f):=\sum_{\alpha\leq p,\beta \leq p}\sup \vert x^\alpha \partial^\beta f(x) \vert\text{ for all }f \in S(\mathbb{R}).$$
What I have done:$$N_p(\psi)=\sum_{\alpha\leq p,\beta \leq p}\sup \vert x^\alpha \partial^\beta\psi(x) \vert$$$$= \sum_{\alpha\leq p,1\leq\beta \leq p}\sup \vert x^\alpha \partial^\beta\psi(x) \vert+\sum_{\alpha\leq p}\sup \vert x^\alpha \psi(x) \vert$$$$=N_p(\phi)-\sum_{\alpha\leq p}\sup \vert x^\alpha \partial^p \phi(x) \vert+\sum_{\alpha\leq p}\sup \vert x^\alpha \psi(x) \vert$$
(I tried to copy it without failing the indexes, but if I did, I tried to isolate $N_p(\phi)$ in the terms using the fact $\psi$ is a primitive of $\phi$)
I think there is something to find here, what I have written may not be what is waited I believe, I mean a common form should be found, I think, what would be the "good way" to Estimate $N_p$ norm of $\psi$ in function of $N_p(\phi)$.