Example 4. Find $\int \frac{d x}{\sqrt{7 x^{2}+9 x+1}}$.
Solution. Let $\sqrt{7 x^{2}+9 x+1}=\sqrt{7} x+t$. Then $7 x^{2}+9 x+$ $+1=7 x^{2}+2 \sqrt{7} x t+t^{2}$

\[
\begin{aligned}
& x=\frac{t^{2}-1}{9-2 \sqrt{7} t}, \quad d x=\frac{-2 \sqrt{7} t^{2}+18 t-2 \sqrt{7}}{9-2 \sqrt{7} t^{2}} d t \\
& \sqrt{7} x+t=\frac{\sqrt{7} t^{2}-\sqrt{7}}{9-2 \sqrt{7} t}+t=\frac{-\sqrt{7} t^{2}+9 t-\sqrt{7}}{9-2 \sqrt{7} t}
\end{aligned}
\]

The given integral takes the form

$-\int \frac{2 d t}{2 \sqrt{7} t-9}=-\frac{1}{\sqrt{7}} \ln |2 \sqrt{7} t-9|+C=$

\[
-\frac{1}{\sqrt{7}} \ln \left|2 \sqrt{7}\left(\sqrt{7 x^{2}+9 x+1}-\sqrt{7} x\right)-9\right|+C
\]