I was trying to evaluate this integral for $n\in \Bbb R:$
$$ I_n=\int_0^n ne^{\frac{1}{\log(x/n)}}~dx$$
But I could not find a way (except for $n=1$). Wolfram Alpha found a closed form for $n=1$ but failed to find any other closed forms.
Then I observed that the integral could be simplified to:
$$I_n=n^2\int_0^1 e^{\frac{1}{\log(x)}}~dx $$
And so $I_n$ is a parabola multiplied by a constant, and a closed form is found.
Is my simplification step valid?
Disclaimer: This answer is short but it is all that is necessary to respond to the question.
Your simplification is correct, which follows from substituting $u=\frac xn$, $dx=ndu$.