If $f$ us periodic and even, what I can conclude about of $\int f \;dx$?

Question

Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a periodic, even and differentiable function. If $L>0$ is the minimal period of $f$, what can I conclude about $$I :=\int_{0}^{L} f(x)\; dx?$$

By the hypotheses we have $$f(0)=f(L) \quad \text{and} \quad f'(0)=0.$$ My intuition tells me that we can conclude that $ I = 0 $. Is this, in general, true?

Answer 1

Take $f(x)=\sin^{2}x$ for a counter-example. Here $L=\pi$ and the integral is strictly positive.

Answer 2

You cannot draw any conclusion about $I$ if $f$ is an even function.

But if $g(x)$ is a periodic odd integrable function and $L$ is the minimum period, then

$$\int\limits_{0}^{L} g(x)\; dx = \frac12 \int\limits_{-L}^{L} g(x)\; dx$$ because $g$ is periodic, while $$\int\limits_{-L}^{L} g(x)\; dx=0$$ because $g$ is odd, so $$\int\limits_{0}^{L} g(x)\; dx = 0.$$