<s>[INST] Does $\int_0^\infty \frac{\cos x}{1+x}$ absolutly converge? 
Does the following indefinite integral converge?
  $\int_0^\infty \frac{\cos x}{1+x}$
converges, absolutely converges?

i can say that by the Dirichlet test it does converge.
i am trying to prove it diverges absolutely (seems close to $\frac{1}{x}$)
unsuccessfully so far
how do i prove /disprove it absolutely converge?
 [/INST] Consider intervals $[2k\pi-\frac{\pi}{4},2k\pi+\frac{\pi}{4}]$ for $k \in \mathbb{N}$, then:
$$\int_{0}^{\infty}\frac{|\cos(x)|}{1+x}dx>\int_{\bigcup_{k \in \mathbb{N}}[2k\pi-\frac{\pi}{4},2k\pi+\frac{\pi}{4}]}\frac{|\cos(x)|}{1+x}dx=\sum_{k=1}^{\infty}\int_{2k\pi-\frac{\pi}{4}}^{2k\pi+\frac{\pi}{4}}\frac{|\cos(x)|}{1+x}dx$$
But on each interval $[2k\pi-\frac{\pi}{4},2k\pi+\frac{\pi}{4}]$ we have $\cos(x)>\frac{1}{2}$, so:
$$\sum_{k=1}^{\infty}\int_{2k\pi-\frac{\pi}{4}}^{2k\pi+\frac{\pi}{4}}\frac{|\cos(x)|}{1+x}dx>\sum_{k=1}^{\infty}\int_{2k\pi-\frac{\pi}{4}}^{2k\pi+\frac{\pi}{4}}\frac{1}{2(1+x)}dx$$
Now we have:
$$\int_{2k\pi-\frac{\pi}{4}}^{2k\pi+\frac{\pi}{4}}\frac{1}{2(1+x)}dx>\int_{2k\pi-\frac{\pi}{4}}^{2k\pi+\frac{\pi}{4}}\frac{1}{2(1+2k\pi+\frac{\pi}{4})}dx=\frac{\pi}{2}\frac{1}{2(1+2k\pi+\frac{\pi}{4})}$$
So:
$$\sum_{k=1}^{\infty}\int_{2k\pi-\frac{\pi}{4}}^{2k\pi+\frac{\pi}{4}}\frac{1}{2(1+x)}dx>\sum_{k=1}^{\infty}\frac{\pi}{2}\frac{1}{2(1+2k\pi+\frac{\pi}{4})}=\infty$$
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