Example 4. Find the integral $\int \sqrt{x} d x$.
Solution.

$$
\int \sqrt{x} d x=\int x^{\frac{1}{2}} d x=\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C=\frac{2}{3} x^{\frac{3}{2}}+C=\frac{2}{3} x \sqrt{x}+C
$$