Periodic function defined by integral

Question

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a differentiable and integrable function of period $T$. It is true that the function $h(t)=\int_{a}^{t} f(\tau) d\tau$ is periodic?

I think it's not true, because for $h$ be periodic we need to find some $Q$ such that $$ \int_{t}^{Q}f(\tau)d\tau = 0 $$

For every $t$, and this is unlike to happen in general. I need help to find a counterexample for the affirmation or a proof for the affirmation.

Answer 1

Let $f(t) = \sin(t) + 1$. Then, $h(t) = \int_{a}^{t} \sin(\tau) + 1\ d\tau = \sin(t)-\sin(a) + x(t - a)$, which is not periodic due to the linear term. $\blacksquare$