How Do You Calculate the Integral of f(x) from 0 to e?

[anemone]

Here is this week's POTW:

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Let $f$ satisfy $x=f(x)e^{f(x)}$. Calculate \(\displaystyle \int_{0}^{e} f(x)\,dx\).

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to lfdahl for his correct solution:), and you can find the suggested solution below:

Spoiler

Note that $f$ is monotonically increasing and is the inverse of the function $g(y)=ye^y$.

Since $f(e)=1$, the area under $f(x)$ from 0 to $e$ is the area of the rectangle with vertices $(0,\,0),\,(e,\,0),\,(0,\,1),\,(e,\,1)$ minus the the area to the left of $f(x)$ from 0 to 1, and the latter is just the integral of $g(y)$ from 0 to 1. So we have

\(\displaystyle \begin{align*}\int_{0}^{e} f(x)\,dx&=e-\int_{0}^{1} g(y)\,dy\\&=e-\int_{0}^{1} ye^y\,dy\\&=e-\left[ye^y\right]_0^1+\int_{0}^{1} e^y\,dy\\&=\int_{0}^{1} e^y\,dy\\&=\left[ye^y\right]_0^1\\&=e-1\end{align*}\)