Evaluate the definite integral $\int_4^8 e^{4-x}dx$

Question

5.4

Evaluate the definite integral $\int_4^8 e^{4-x}dx$

Can somebody verify this solution for me?? Thanks!

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So we could evalute this with $u$ substitution, by letting $u=4-x$ and proceeding like that... But instead I'm going to do something else:

$\int_4^8 e^{4-x}dx$

$=\int_4^8 e^{4}e^{-x}dx$

$= e^{4}\int_4^8e^{-x}dx$

$= e^{4}(\frac{e^{-x}}{-1}|_4^8)$

$= -e^{4}(e^{-x}|_4^8)$

$= -e^{4}(e^{-8}-e^{-4})$

$= (-e^{4})e^{-8}-(-e^{4})e^{-4})$

$= -e^{-4}+1$

Answer 1

It should be $(-e^4)e^{-8}-(-e^4)e^{-4}=\color{red}-e^{-4}+1$.