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I was tasked with finding the integral:

$$\int_{0}^{1}\int_{e^x}^{e}{\frac{dydx}{\log{y}}}$$

First, I tried to find the integral inside first, which is:

$$\int_{e^x}^{e}{\frac{dy}{\log{y}}}$$

I substituted $e^t=x$ and went on to get:

$$\int_{t}^{1}{\frac{e^tdt}{t}}$$

Now, I tried plugging this into the integral calculator, but unfortunately, an antiderivative doesn't exist in terms of known functions.

Right now I'm stuck and really don't know how to proceed. Can someone guide me to along the right path?

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  • $\begingroup$ First a few errors, you mean $e^t = y$ as substitution I think? The last integral must be $\int_t^1 \frac{e^t}{t} dt$. $\endgroup$ – Tobias Molenaar Nov 27 '17 at 13:15
  • $\begingroup$ Yes yes, $e^t=y$ it is $\endgroup$ – Pritt Balagopal Nov 27 '17 at 13:18
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    $\begingroup$ Change the order of integration: $\endgroup$ – user247327 Nov 27 '17 at 13:18
  • $\begingroup$ And yes, thats a mistake too. @TobiasMolenaar $\endgroup$ – Pritt Balagopal Nov 27 '17 at 13:20
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Reverse the order of integration.

\begin{equation} \begin{split} \int_0^1 \int_{e^x}^e \frac{\ dy\ dx}{\ln y } &= \int_1^e\int_0^{\ln y}\frac{\ dx\ dy}{\ln y }\\ &= \int_1^e 1 \ dy\\ &= e- 1. \end{split} \end{equation}

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