$\int \left[\left(\frac{x}{e}\right)^x + \left(\frac{e}{x}\right)^x\right]\ln x \,dx$ Integrate:
$$\int \left[\left(\frac{x}{e}\right)^x + \left(\frac{e}{x}\right)^x\right]\ln x \,dx$$
This question looks like of the form of $$\int\ e^x(f(x)+f'(x))\,dx,$$ but don't know how to get the proper substitution?

Hint
Try with
\begin{align*}
y & = \left(\frac{e}{x}\right)^{x}\\
\ln y & = x[1-\ln x]\\
\frac{y^{'}}{y} & = -\ln x.
\end{align*}
Now for the first component think of a similar idea.
