TITLE: The integral of $\left|\frac{\cos x}x\right|$
QUESTION [7 upvotes]: I'm looking to determine whether the following function is unbounded or not:
$$
F(x) = \int_1^x\left|\frac{\cos t}{t}\right|\text{d} t
$$
I can't seem to do much with it because of the $|\cos(t)|$. I thought of using the fact that $\int |f| \ge |\int f|$, but the problem is that the integral of $\frac{\cos t}t$ (without the absolute values) is bounded, and so that doesn't prove that $F(x)$ is unbounded or bounded. I tried re-expressing this as a cosine integral (the function $\text{Ci}(x)$) but to no avail. I'm not sure where else to go with this; the main problem seems to be the fact that its very difficult to derive an inequality with the $|\cos(t)|$ without a $|\cos(t)|$ on the other side of the inequality (or at least some trig function).
Any help would be appreciated.

REPLY [5 votes]: Hint:  
Consider the harmonic series and 
$$\int_{\pi/2 + k\pi}^{3\pi/2 + k\pi} \frac{| \cos t|}{t} \, dt \geqslant \frac{1}{3\pi/2 + k \pi}\int_{\pi/2 + k\pi}^{3\pi/2 + k\pi} |\cos t| \, dt  = \frac{2}{3\pi/2 + k \pi}$$