I've been struggling for a while now on evaluating this disgusting integral:
$$(\ln2)\int_0^{(\ln2)^{1/\ln2}}2^{\ln x}\cdot x^\left(\frac{x^{\ln2}+1}{\ln x}-1\right)dx$$
My maths teacher gave our class this question a while ago, and he said that we should be able to do it (I am in high-school, and we have only been taught a fairly basic level of integration).
So today I spent many hours applying every integration technique I know to this monster, but I got absolutely nowhere. It got to a point where I couldn't think of another variable to use as a substitution because I had already made so many.
I eventually decided to plug this into an integral calculator and received a surprisingly nice result of $e$, however there was no further information and so I was not able to view any of the steps in how they got there.
I am so stuck on this problem :(
Does anyone know how they got there? What are the steps in finding its indefinite form?