**Bug introduced in 7.0 or earlier and persisting through 10.3.1 or later** --- *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 $10^{114}$