Good evening,
I'm studying integrals at the moment, and right now I'm studying a part where our teacher shows us how to see if an integral exists (is not infinite or 0) by comparing to the integral of a max or min integral.
For example in order to solve this:
$\int_2^\infty\frac{\cos{x}}{x^2}dx$
the example shows that the main part of the integral is equal and less than $\frac{1}{x^2}$ for $x\in[2, +\infty)$ so
$0 \le \frac{|\cos{x}|}{x^2} \le \frac{1}{x^2}$
thus saying that the $\int_2^\infty\frac{1}{x^2}dx$ exists (and is $\frac{1}{2}$) so the starting integral exists as well.
This makes perfect sense to me, and I can see how $\frac{1}{x^2}$ is equal or larger than $\frac{|\cos{x}|}{x^2}$.
However, what baffles me is the next 2 examples.
a) $\int_1^\infty\frac{3x^2+2x+1}{x^3+6x^2+x+4}dx$
b) $\int_1^\infty\frac{3x+1}{\sqrt{x^5+4x^3+5x}}dx$
For which he says that for
a) $\frac{3x^2+2x+1}{x^3+6x^2+x+4} \ge \frac{1}{4x}, x\in[1,+\infty)$
b) $0 \le\frac{3x+1}{\sqrt{x^5+4x^3+5x}} \le \frac{4}{x^{3/2}}$
What I can't understand or find any way to see how, is how he came up with $\frac{1}{4x}$ and $\frac{4}{x^{3/2}}$.
Is this a specific routine I have to follow somehow to find those max and min functions?
Thanks in advance for any help and explanation.