Example 3. Calculate $\int_{4}^{9}\left(\frac{2 x}{5}+\frac{1}{2 \sqrt{x}}\right) d x$.
Solution. Based on properties 5 and 6 of the definite integral and formula (5.3), we obtain

$$
\begin{aligned}
& \int_{4}^{9}\left(\frac{2 x}{5}+\frac{1}{2 \sqrt{x}}\right) d x=\frac{2}{5} \int_{4}^{9} x d x+\int_{4}^{9} \frac{1}{2 \sqrt{x}} d x=\left.\frac{2}{5} \cdot \frac{x^{2}}{2}\right|_{4} ^{9}+ \\
& +\left.\sqrt{x}\right|_{4} ^{9}=\frac{1}{5}\left(9^{2}-4^{2}\right)+(\sqrt{9}-\sqrt{4})=\frac{1}{5} \cdot 65+1=14
\end{aligned}
$$