lol math competition

Okay, can't do these two.

Prove sum(k=1,n){arctan(1/(1+k+k^2))} = arctan(n/(n+2))

Show integral(0,infinity) e^{(x^2 + x^-2)/4} = sqrt(pi/e)

>>3
My 89 says that it's ∞, too. I'd normally trust Mathematica over my calculator, but Mathematica's being a dick tonight—it's spitting out some bullshit long answer, and refuses to simplify or even approximate it. And hell if I'm going to integrate this late at night.

So, are you sure it's √π/e?