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

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

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

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  • $\begingroup$ What is the motivation for the question? The boundary condition for $f'$ is not very helpful when doing integration by parts. $\endgroup$ – Jack Dec 19 '16 at 16:47
  • $\begingroup$ 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. $\endgroup$ – gibarian Dec 19 '16 at 16:48
  • $\begingroup$ Then what is the point for the question? $\endgroup$ – Jack Dec 19 '16 at 16:49
  • $\begingroup$ 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. $\endgroup$ – gibarian Dec 19 '16 at 16:50
  • $\begingroup$ Not true. Consider $f'(x)=x(x-\pi)$. $\endgroup$ – Jack Dec 19 '16 at 16:54
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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, $T(V)= \mathbb R$

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