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Solve the integral Cos(x)/x at the interval (0,infinite)

Dear All,

I am trying to find Integral[Cos[x]/x,{x,0,infinite}]. Integrate can not give a result since the expression is divergent. However, NIntegrate gives a result that I do not trust.

Integrate[1/x Cos[x], {x, 0, \[Infinity]}]
NIntegrate[1/x Cos[x], {x, 0, \[Infinity]}]

Does anyone know if this integral can be taken? And how can we trust the result?

Best wishes,

İsa
POSTED BY: Isa Comez
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The value of the integral is Infinity, in this sense:

Integrate[Cos[x]/x, {x, a, b}]
Limit[%, {a -> 0, b -> Infinity},
 Direction -> {"FromAbove", "FromBelow"}]
POSTED BY: Gianluca Gorni
alis sac
alis sac
Posted 10 hours ago

The integral is conditionally convergent in the improper/Cauchy principal-value sense, and the result can be verified using the cosine integral function rather than relying blindly on NIntegrate.
POSTED BY: alis sac

It is somewhat disappointing that NIntegrate does not decide at once that the integral diverges.
POSTED BY: Gianluca Gorni 
 int = Integrate[Cos[x]/(x + e), {x, 0, \[Infinity]}, Assumptions -> e >= 0]
 Limit[int, e -> 0]

(* \[Infinity] *)
POSTED BY: Mariusz Iwaniuk

The integral diverges because of the singularity at x==0. NIntegrate does not commit, but it gives serious warnings. Compare with converging integrals:

Integrate[1/x Sin[x], {x, 0, \[Infinity]}]
NIntegrate[1/x Sin[x], {x, 0, \[Infinity]}]
POSTED BY: Gianluca Gorni
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