*Mathematica* gives the following very odd result,

    Integrate[Sin[Sqrt[x]]/Sqrt[x], {x, 1, ∞}]
    (* 2 Cos[1] *)

which seems unintuitive.  The integrand has an easy to find antiderivative,

    Integrate[Sin[Sqrt[x]]/Sqrt[x], x]
    (* -2 Cos[Sqrt[x]] *)

When we evaluate this at the limits of integration, nothing surprising,
    
    Function[x, -2 Cos[Sqrt[x]]] /@ {∞, 1}
    (* {Interval[{-2, 2}], -2 Cos[1]} *)

And it isn't that *Mathematica* can't handle the difference between an `Interval` object and a number,

    Differences@%
    (* {Interval[{-2 - 2 Cos[1], 2 - 2 Cos[1]}]} *)

So why does it seem so confident that the answer is `2 Cos[1]`?