$f\colon [0,+\infty)\to \mathbb{R}$ is an integrable function.
$$g(x) := \int_0^x \frac{f(t)}{1+t^2} \,dt$$
I known that since $f$ is integrable, its indefinite integral is continuous by the Fundamental Theorem of Integral Calculus. But I don't think the composite of integrable functions is integrable ($f$ and $\frac{1}{1+x^2}$), and I need this to complete a step in a proof.