Integral knowing initial conditions for the derivative function.

I would like to know what can we know about the following integral:

$$\intop_{0}^{\pi}f'(x)\cos x\ dx$$

whenever $f'(0)=f'(\pi)=0$.

• What is the motivation for the question? The boundary condition for $f'$ is not very helpful when doing integration by parts. – Jack Dec 19 '16 at 16:47
• Yes. The problem is that know nothing about $f(0)$ and $f(\pi)$. On the other hand I recover the initial integral when doing parts twice. – gibarian Dec 19 '16 at 16:48
• Then what is the point for the question? – Jack Dec 19 '16 at 16:49
• I suspect the integral is zero (being cosinus a pair function) but I don't know if there is a bounding or solving theorem about these kind of integrals. It comes from a problem of inflating spherical balloons. – gibarian Dec 19 '16 at 16:50
• Not true. Consider $f'(x)=x(x-\pi)$. – Jack Dec 19 '16 at 16:54

Let us consider the vector space $V=\{f \in C1[0, \pi]: f'(0)=f'( \pi)=0\}$ and the linear functional $T:V \to \mathbb R$, defined by
$T(f)=\intop_{0}^{\pi}f'(x)\cos x\ dx$.
Since $T$ is not the zero-functional, $T(V)= \mathbb R$