How do you solve $\int_{0}^{1}\int_{e^x}^{e}{\frac{dydx}{\log{y}}}$?

Question

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?

Answer 1

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}