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The answer in the book is $e-1$, and but I can't figure out how to go from $x(\ln(x) - 1)$ to that answer...

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Your area is nothing but $\displaystyle \int_{y=0}^{y=1} x dy$ where $x = e^y$. Plug this in to get the answer.

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We have the following:

$$ \int_{y=0}^{y=1} e^y \ dy$$ which is

$$e-1$$

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