Question

How do you determine if the improper integral converges or diverges `int 8dx/(x^(2)+1)` from 1 to infinity?

Answer 1

I assume (despite the 8) you want to know if `lim_(a to oo)int_1^adx/(1+x^2)` exists and, if so, what its value is. The limit exists and is `pi/4`.

Explanation:

The integral is a standard integral having the value `tan^-1 x +C`.
So the definite integral has the value `tan^-1 a - tan^-1 1` which is `tan^-1 a -pi/4`. As `a to oo,` `tan^-1 a to pi/2`. (The shorthand for the limit is `int_1^oo dx/(1+x^2)`.)