I am interested in a proof of the following.
$ \int_1^{\infty} \dfrac{\{t\} (\{t\} - 1)}{t^2} dt = \log \left(\dfrac{2 \pi}{e^2}\right)$ where $\{t\}$ is the fractional part of $t$.
I obtained a circuitous proof for the above integral. I'm curious about other ways to prove the above identity. So I thought I will post here and look at others suggestion and answers.
I am particularly interested in different ways to go about proving the above.
I'll hold off from posting my proof for sometime to see what all different proofs I get for this.