The integral of the function $\frac{1}{9x^2 + 16x + 5}$ with respect to $x$ is:

$\int \frac{1}{9x^2 + 16x + 5} \, dx = \frac{\sqrt{19}}{38} \ln\left|x - \frac{\sqrt{19}}{9} + \frac{8}{9}\right| - \frac{\sqrt{19}}{38} \ln\left|x + \frac{\sqrt{19}}{9} + \frac{8}{9}\right| + C$

where $C$ is the constant of integration.

Would you like further details or have any questions? Here are five related questions:

1. How do you solve integrals involving quadratic expressions in the denominator?
2. Can you explain the method of completing the square in relation to integrals?
3. How is the logarithm function derived in the context of integrals like this one?
4. What are some examples of integrals involving more complex quadratic expressions?
5. How does the constant of integration affect the final answer?

Tip: When integrating functions with quadratic expressions, completing the square can simplify the process and make the integral easier to solve.