Is it true that \begin{align} \int_0^A (A-x) \cos(x) \ f(x) dx \ge 0 \end{align} for all $A\ge 0$, if for $f(x)$ monotone decreasing, non-negative and bounded.
This question came up in the discussion here where it was also shown that for $A \le \pi$ this is true.
Example:
For $f(x)=e^{-tx}$ we have that
\begin{align} \int_0^a (a-x) \cos (x) e^{-t x} \, dx=\frac{e^{-a t} \left(\left(t^2-1\right) \cos (a)-2 t \sin (a)+e^{a t} \left(a \left(t^3+t\right)-t^2+1\right)\right)}{\left(t^2+1\right)^2} \end{align}
the expression $\left(t^2-1\right) \cos (a)-2 t \sin (a)+e^{a t} \left(a \left(t^3+t\right)-t^2+1\right)$ apprears to be postive for all $a>0$ and $t>0$.