Determine whether the integral converges or diverges. Indicate on a graph the region whose weighted area is given by the integral. If the integral converges, find its value.
We have to split up the integral into two improper integrals with only one badly-behaved limit of integration each:
Evaluate the first improper integral:
Looking at a graph of the function arctan x we can see that as b approaches -∞, arctan b approaches :
Now for the other integral. We look at the graph of arctan x to figure out what arctan b approaches as b approaches ∞.
Putting these results together,
It's a good sign that we got a positive number, since the area between the graph of and the x-axis all lies on top of the x-axis.