Is the following integral convergent?

Question

Is $\int_0^{\infty}(1-e^{-x})\frac{\cos x}{x}$ dx convergent? Any help will be very helpful. I tried to use comparison test with bounding $\cos x$ by 1 and $(1-e^{-x})$ by 1 and so on but the series to which i tried to compare became divergent so I was unable to conclude any further.

Answer 1

Near zero, the integrand is $$(x+O(x^2))\frac{\cos x}x$$ so the integral converges there.

At infinity the integral of $x^{-1}\cos x$ converges conditionally (compare $x^{-1}\sin x$) and that of $e^{-x}x^{-1}\cos x$ converges absolutely.

The answer is then, yes.