Gronwall inequality for integrable functions

Question

In the book "Some Gronwall type inequalities and applications" (Theorem 5): Let $x:[a,b] \to \mathbb{R}$ a continuous function that satisfies the inequality $$\frac{x^2(t)}{2} \leq \frac{x^2_0}{2}+\int_a^t \Psi (s)x(s) ds, \ t \in [a,b]$$ where $x_0 \in \mathbb{R}$ and $\Psi$ are nonnegatives continous in $[a,b]$. Then the estimation $$|x(t)|\leq |x_0|+\int_a^t\Psi(s)ds, \ t \in [a,b]$$ holds. Can we to have the same result if replace "$x$ continuous" for "$x$ integrable" or $x \in L^2(0,\infty; \mathbb{R})$ (for example)? Im trying addapt the solution for $x$ integrable. I appreciate any help.