Say we have a function $f:[a,b] \to \mathbb{R}$ and we assume that this function is Riemann integrable. This implies that $f$ is bounded, say by $M$. Then $|f| \le M$.
Now consider $F:[a,b] \to \mathbb{R}$ where $F(x) = \int_{a}^{x}f(t)dt$. Because $f$ is bounded by $M$ do we have that $F(x)$ is bounded by $M(x-a)$? I feel this result might not be true, but there is a similar result for bounding an integral, does anyone have any ideas?
Let $|f| \le M$ then
$$ \int_a^x |f(y)|dy \le \int_a^x Mdy = M\int_a^x dy = M(x-a)$$
and $$\left \lvert \int_a^x f(y)dy \right \rvert \le \int_a^x |f(y)|dy.$$
So what can we conclude?
Remark: You have written $\int_a^xf(x)dx$ but I am taking the assumption you didn't mean to have the same variable as your upper limit of integration as well as the variable you are integrating with respect to.