$f:[0,1]\rightarrow \mathbb{R}$ be continuous, then $ \int_{0}^{1}f(x)e^{-x}dx=?$
I understand I need to do integration by parts, taking $f(x)$ in 2nd function, so that I can apply mean value theorem of Integral calculus $\int_{0}^{1}f(x)=(1-0)f(c)$ for some $c\in(0,1)$, but I am not able to get the final answer
will the answer b from the following
$f'(0)-f'(1)e^{-1}$
$f(c)(1-e^{-1})$
$e^{-c}\int_{0}^{1}f(x)dx$