is it possible to integrate something like $\displaystyle e^x.\frac{1}{2\sqrt{x}}$
because neither part will cancel so i just keep getting stuck in a perpetual loop
Note that : $\displaystyle e^x=e^{\left( \sqrt{x} \right)^2}$
If you substitute $\displaystyle u=\sqrt{x}$, you will get :
$\displaystyle \int e^{u^2} \, du$ , which is well-known unelementary integral ..
Your integral can not be expressed in terms of elemenetary integrals ..