Originally Posted by

**MathoMan** No you dont use substitution.

You should recall that: $\displaystyle df(x)=f'(x)dx$ and if $\displaystyle f(x)=\sqrt{x}$ you actually have $\displaystyle d(\sqrt{x})=\frac{1}{2\sqrt{x}}dx$ which you can rewrite and use in that integral in the form of $\displaystyle \frac{dx}{\sqrt{x}}=2d(\sqrt{x})$.

$\displaystyle \int{\frac{sin(\sqrt{x})}{\sqrt{x}}}dx=\int\sin{(\ sqrt{x})}\frac{dx}{\sqrt{x}}=\int\sin (\sqrt{x})2d(\sqrt{x})=2\int\sin (\sqrt{x})d(\sqrt{x})=-2\cos(\sqrt{x})+C.$

You should always bear in mind that you could (and should be able to) use differential calculus just as you would use algebraic manipulation in order to simplify the integral.