Improper Integrals - Creating a function for the comparison tests

Mar 2020
Buenos Aires
When it comes to evaluating if an improper integral converges or diverges and it's not possible to find it's primitive, using the comparison tests becomes very useful (I'm specifically talking about the direct comparison test and the limit comparison test). And for that, you've got to come up with a new function in order to see how the original one behave. Speaking for me, this has been my biggest obstacle until now. So here I am asking you, superior beings, for some advice and also, due to the lack of information I experience on how to face this particular challenge, if you could share me some books, links or even class notes that you know of or have about the subject. I really appreciate your effort and thank you for your time in advance!
Dec 2015
from class notes : show whether integral diverges or not ? \(\displaystyle I=\int \limits_{0}^{\infty} \frac{x\sin(x) }{x^2 +x +1} dx .\)