Example 1. Find the integral $\int \frac{\sqrt{x+9}}{x} d x$.
Solution. This is an integral of the form (4.32), for which $\frac{a x+b}{c x+d}=x+9$ (i.e., $a=1, b=9, c=0, d=1$), $\frac{p_{1}}{q_{1}}=\frac{1}{2}$.

We apply the substitution

$$
x+9=t^{2}
$$

from which

$$
x=t^{2}-9, d x=2 t d t
$$

and

$$
\begin{gathered}
\int \frac{\sqrt{x+9}}{x} d x=\int \frac{t}{t^{2}-9} 2 t d t=2 \int \frac{t^{2}}{t^{2}-9} d t=2 \int \frac{\left(t^{2}-9\right)+9}{t^{2}-9} d t= \\
=2 \int d t+18 \int \frac{d t}{t^{2}-9}=2 t+\frac{18}{2 \cdot 3} \ln \left|\frac{t-3}{t+3}\right|+C= \\
=2 \sqrt{x+9}+3 \ln \left|\frac{\sqrt{x+9}-3}{\sqrt{x+9}+3}\right|+C
\end{gathered}
$$