Convergence of an integral under a condition [closed]
Let $F:\mathbb R\to\mathbb R$ such that \begin{align} \int_0^\infty F(x)e^{-x}dx<\infty. \end{align}
Is it true that
$$\int_0^\infty F(x)e^{-x}xdx<\infty?$$
You can manipulate $F(x)$ to your heart's desire, so your question is equivalent to the following: given that $\int_0^\infty f(x) dx < \infty$, do we always have $\int_0^\infty xf(x)dx < \infty$?
The answer is obviously no, can you pick some $f(x) = 1/(x-1)^k$ for example for the right value of $k$?