Infinite integral of $1/(1+x^2)$

Question

Given the theorem that the infinite integral of $1/x^n$ is convergent if and only if $n>1$, I want to prove that the infinite integral of $1/(1+x^2)$ exists. This seems like a trivial question, I know, and one can calculate the integral by other means but I'd like to understand the argument from a book I'm reading.

Answer 1

Split up the integral into two pieces: $$ \int_{0}^{\infty}\frac{dx}{1+x^2}=\int_{0}^1\frac{dx}{1+x^2}+\int_1^{\infty}\frac{dx}{1+x^2} $$ The first integral presents no problems, and for the second integral we have $$ \int_1^{\infty}\frac{dx}{1+x^2}\leq \int_1^{\infty}\frac{dx}{x^2}<\infty $$ since $\frac{1}{1+x^2}\leq \frac{1}{x^2}$ on $[1,\infty)$.