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$f:[0,1]\rightarrow \mathbb{R}$ 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

  1. $f'(0)-f'(1)e^{-1}$

  2. $f(c)(1-e^{-1})$

  3. $e^{-c}\int_{0}^{1}f(x)dx$

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    Could you bit a little more precise in asking your question? I can't tell what you want here.2012-07-28

1 Answers 1