What is ∫(sin(√(x))/x)dx?
The best we can do is to let $\displaystyle u = \sqrt{x} \implies 2u~du = dx$, so
$\displaystyle \int \frac{sin(\sqrt{x})}{x}~dx = \int \frac{sin(u)}{u^2} \cdot 2u~du = 2 \int \frac{sin(u)}{u}~du$
This integral (at least) sometimes goes by the name of "Si." (So as a function of x it is $\displaystyle Si(x)$.) There is no closed-form solution to this integral.
-Dan