How would you apply complex analysis/the residue theorem to evaluate the following improper integral:
$\displaystyle
\int_{0}^{\infty} \frac{2 t}{\left(t^{2}+1\right)^{2}} \,dt$
Do you really have to pick an integral which can easily be solved with $\displaystyle x=t^2+1$?
It it just Palinful to do an integral which solves easily with substitution.
Is it okay with you if I do a less trivial integral such as, $\displaystyle \int_0^{\infty} \frac{dt}{(t^2+1)^2}$?