00:01
So in this question, we want to determine whether the integral from negative infinity to positive infinity of 2 over 4 plus x squared dx converges or diverges.
00:11
So what am i going to do? well, i'm going to start, since this is a doubly improper integral, by splitting this at x equals 0.
00:21
I'm going to say this is the integral from negative infinity to 0 of 2 over 4 plus x squared dx.
00:33
Plus the integral from 0 to infinity of 2 over 4 plus x squared dx.
00:43
Now, once i do that, i'm going to rewrite each of these integrals as limits.
00:49
So my first integral, i can write that as the limit as a approaches negative infinity of the integral from a to 0 of 2 over 4 plus x squared dx.
01:09
While my second integral, i can write that as the limit as b approaches infinity of the integral from 0 to b of 2 over 4 plus x squared dx.
01:27
Now, i want to recall how to integrate something like 2 over 4 plus x squared.
01:36
It is a fact that if you're integrating 1 over a squared plus x squared dx, this is equal to, 1 over a times the arc tangent of x over a plus c and so if i'm trying to find an anti -derivative of 2 over 4 plus x squared d x well i'm thinking of that 4 as 2 squared right so what am i getting for this anti -derivative well i've got the 2 coming along for the ride from here times 1 over 2 2 times the arc tangent of x over 2.
02:38
Okay.
02:40
And yes, there is a plus c, although i won't need it when i go to my definite integral.
02:46
So my anti -derivative here is just the arc tangent of x over 2...