How do you show whether the improper integral #int 1/ (1+x^2) dx# converges or diverges from negative infinity to infinity?
1 Answer
I would prove that it converges by evaluating it.
Explanation:
If you don't know, or have forgotten the "formula", then use a trigonometric substitution:
Recall that
We need to split the integral.
# = lim_(ararr-oo) [tan^-1 x]_a^0 +# #lim_(brarroo) [tan^-1 x]_0^b#
# = lim_(ararr-oo)[tan^-1(0) - tan^-1(b)] + lim_(brarroo)[tan^-1b-tan^-1 0]#
# = lim_(ararr-oo)[0 - tan^-1(b)] + lim_(brarroo)[tan^-1b-0]#
# = [-(-pi/2)]+[pi/2] = pi#
The integral converges.