Evaluate $\int \frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x}}\,dx$.

Explanation

1. 01

   Put $t = \sqrt{x}$; then $dt$ is the rest of the integrand, up to a constant.

2. 02

   The integral becomes a standard form in $t$; putting $t = \sqrt{x}$ back gives $2 \sin{\left(\sqrt{x} \right)} + C$.

3. 03

   Check: differentiating the answer gives back the integrand.