I'm once again stuck on checking the convergence of an improper integral. The given question is as follows:
Show that: $$\frac{3}{4}\le \int_{1}^{\infty}\frac{dx}{x^{2}+\left|\cos{x}\right|}\le1$$
I understand why the integral is $\le1$, as I used the fact that $\frac{1}{x^{2}+\left|\cos{x}\right|}\le\frac{1}{x^2}$. From this I could use the comparison test to conclude that the integral has to be $\le1$.
However, I can't really figure out how to conclude the fact that the integral should be $\ge\frac{3}{4}$. Any advice would be appreciated!
You have\begin{align}\int_1^\infty\frac1{x^2+|\cos x|}\,\mathrm dx&\geqslant\int_1^\infty\frac1{x^2+1}\,\mathrm dx\\&=\left(\lim_{x\to\infty}\arctan x\right)-\arctan1\\&=\frac\pi2-\frac\pi4\\&=\frac\pi4\\&>\frac34.\end{align}