If $\int_a^x f(t)\, dt$ is differentiable, is its derivative integrable?

Question

Let $f$ be a real-valued Lebesgue integrable function on $[a,b]$. If $F(x) = \int_a^x f(t)\, dt$ is differentiable on $[a,b]$, is $F'(x)$ (Lebesgue) integrable?

I know there are examples of differentiable functions whose derivative is not Lebesgue integrable, for example $F(x) = x^2\sin(1/x^2)$ on $[0,1]$, $F(0) = 0$. But I do not think this can arise as an integral.

Answer 1

According to Lebesgue's Differentiation Theorem you have that $F'(x)=f(x)$ a.e. and you know $f$ is integrable, so...