Question

find the definite integral that is equivalent to lim n->infinity of sum over i=(1,n) n/(n^2+i^2)

since definite integral of f(x) dx over (a,b) = lim n->infinity sum over i=(1,n) f(a + (b-a)i/n) * (b-a)/n

then: f(a + (b-a)i/n) * (b-a)/n = n/(n^2+i^2)
rearrange to: so f(a + (b-a)i/n) = 1/(b-a) * n^2/(n^2 + i^2)

if (b-a)=1, then this simplifies to: f(a + i/n) = n^2/(n^2 + i^2)
I'd guess a=1, so that: f((n + i)/n) = n^2/(n^2 + i^2)

This almost works where f(x) = 1/x^2, but not quite. I'm not sure what else to do here.