PF PotW Robot
Here is this week's intermediate math problem of the week. We have several members who will check solutions, but we also welcome the community in general to step in. We also encourage finding different methods to the solution. If one has been found, see if there is another way. Occasionally there will be prizes for extraordinary or clever methods. Spoiler tags are optional.
Evaluate the integral ##\displaystyle \int_0^{\infty}\frac{dx}{(1+x^2)^{\alpha/2}}## for ##\alpha>1##.
Express your answer using Gamma functions, where
##\Gamma(x) :=\int_{0}^{\infty}t^{x-1}e^{-t} \, dt.##
(PotW thanks to our friends at http://www.mathhelpboards.com/)
Evaluate the integral ##\displaystyle \int_0^{\infty}\frac{dx}{(1+x^2)^{\alpha/2}}## for ##\alpha>1##.
Express your answer using Gamma functions, where
##\Gamma(x) :=\int_{0}^{\infty}t^{x-1}e^{-t} \, dt.##
(PotW thanks to our friends at http://www.mathhelpboards.com/)