`underset(nrarroo)lim[(n+1)/(n^(2)+1^(2))+(n+2)/(n^(2)+2^(2))+(n+3)/(n^(2)+3^(2))+...+(1)/(n)]`

Explanation

`underset(nrarroo)(Lt)underset(r=1)overset(n)sum(n+r)/(n^(2)+r^(2))=underset(nrarroo)(Lt)underset(r=1)overset(n)sum1/n(1+(r)/(n))/(1+(r/n)^(2)) :' underset(nrarroo)(Lt) (r/n) = 0`, when `r = 1`, lower limit = 0
and `underset(nrarroo)(Lt)(r/n) = underset(nrarroo)(Lt)(n/n) = 1`, when r = n, upper limit `= 1`
`underset(0)overset(1)int(1+x)/(1+x^(2))dx = underset(0)overset(1)int(1)/(1+x^(2))dx + 1/2underset(0)overset(1)int(2x)/(1+x^(2))dx = [tan^(-1)x]_(0)^(1)+ [1/2log_(e)(1+x^(2))]_(0)^(1)`
`= (pi)/(4) + 1/2 ln 2`