**Bug introduced in 7.0 or earlier and persisting through 10.3.1 or later**

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*Mathematica* 10 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]`?

**Edit**

We can even go further to see that this is wrong by making the substitution $u^2=x$ ($2u \mathrm{d}u=\mathrm{d}x$)

    Integrate[2*u*(Sin[u]/u), {u, 1, Infinity}]
>    During evaluation of Integrate::idiv:Integral of Sin[u] does not converge on {1,∞}. >>
 
    (* Integrate[2*Sin[u], {u, 1, Infinity}] *)

And just to drive this home even further, we try with `NIntegrate`, which after spitting out error messages gives an answer on the order of $10^{114}$