1. integral -oo, oo [(2x)/(x^2+1)^2] dx

Question

2. integral 0, pi/2 cot(theta) d(theta)

(a) state why the integral is improper or involves improper integral
(b) determine whether the integral converges or diverges
converges?
(c) evaluate the integral if it converges

CONFUSE: how would I know the integral converges or diverges without
evaluating the integral? I am not sure if I evaluated the integral correctly.

(a) infinite limit of integration
(b) diverges?
lim -1/(x^2+1) -oo, b = oo?
b->oo

(a) integrand has an infinity discontinuity at x=0
(b) diverges?
lim cos(theta)/sin(theta) d(theta)
b->oo
= lim ln(sin(theta)) d(theta) b, pi/2
b->oo
=-oo?