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?

Hint. For the convergence, you have, for $M>0$, integrating by parts:
$$
\int_0^M \frac{\cos x}{x+1}dx=\left. \frac{\sin x}{x+1}\right|_0^M+\int_0^M \frac{\sin x}{(x+1)^2}dx
$$ since
$$
\left|\int_0^M \frac{\sin x}{(x+1)^2}dx\right|\leq\int_0^{+\infty} \frac{1}{(x+1)^2}dx<+\infty
$$
giving that your initial integral is convergent. 
The integral is not absolutely convergent, as showned by Jack d'Aurizio.
