Prove $\int\limits_0^{\infty}\left | \frac{\cos(x)}{1+x}\right |dx$ diverges 
Prove $\int\limits_0^{\infty}\left | \frac{\cos(x)}{1+x}\right |dx$ diverges

I tried using $\left | \frac{\cos(x)}{1+x} \right | \geq \frac{\cos^2(x)}{1+x} = \frac{\cos(2x) +1}{2x+2}$ and showing the right side diverges but it got so complicated...
Tips appreciated
 A: HINTS:
Write the integral as the sum
$$\begin{align}
\int_{\pi/2}^\infty \frac{\cos(x)}{1+x}\,dx&=\sum_{k=1}^\infty \int_{(2k-1)\pi/2}^{(2k+1)\pi/2}\frac{|\cos(x)|}{1+x}\,dx\\\\
&=\sum_{k=1}^\infty \int_{\pi/2}^{3\pi/2}\frac{-\cos(x)}{1+x+k\pi}\,dx\\\\
\end{align}$$
Then, note that $\frac1{1+{(k+3/2)\pi}}\le \frac{1}{1+x+k\pi}\le \frac{1}{1+(k+1/2)\pi}$ when $x\in[\pi/2,3\pi/2]$ and that $\int_{\pi/2}^{3\pi/2}(-\cos(x))\,dx=2$.  
A: Break it up into a sum of easier integrals:
$$
\int_0^\infty \left| \frac{\cos(x)}{x+1} \right|\mathrm dx=
\int_0^\frac{\pi}{2}\frac{\cos(x)}{x+1}\mathrm dx+\sum_{k=0}^{\infty}\int_{(2k+1)\frac{\pi}{2}}^{(2k+3)\frac{\pi}{2}}\frac{|\cos(x)|}{x+1}\mathrm dx\\
\geq \sum_{k=0}^\infty\frac{1}{(4k+1)\frac{\pi}{2}+1}\int_{(4k-1)\frac{\pi}{2}}^{(4k+1)\frac{\pi}{2}} \cos(x)\mathrm dx\\
=\sum_{k=0}^\infty\frac{2}{(4k+1)\frac{\pi}{2}+1}
$$
A: we have that:
$$0\le|\cos(x)|\le1$$
but we can consider that the average of this function is:
$$\frac{2}{\pi}\int_0^{\pi/2}\cos(x)dx>0$$
and so:
$$\int_0^\infty\left|\frac{\cos(x)}{1+x}\right|dx\ge a\int_0^\infty\frac 1{1+x}dx\to\infty$$
and so it diverges
A: Hint : let F(x) = integrate( |cos(t)| )dt from 0 to x
Then use integration by parts.
Now employ the following 
If n Pi < x < (n+1) Pi , then
n < F(x) < n+1
Use similar bound for 1/(1+x)^2 .
Now take n to infinity
A: $\begin{array}\\
\int\limits_0^{\infty}\left | \frac{\cos(x)}{1+x}\right |dx
&=\lim_{n \to \infty}\sum_{k=0}^n \int\limits_{k\pi}^{(k+1)\pi}\left | \frac{\cos(x)}{1+x}\right |dx\\
&\gt\lim_{n \to \infty}\sum_{k=0}^n \dfrac1{(k+1)\pi}\int\limits_{k\pi}^{(k+1)\pi}\left | \cos(x)\right |dx\\
&=\lim_{n \to \infty}\sum_{k=0}^n \dfrac1{(k+1)\pi}(2)\\
&=\lim_{n \to \infty}\sum_{k=0}^n \dfrac{2}{(k+1)\pi}\\
&\to \infty\\
\end{array}
$
