We are tasked with evaluating the integral:

$I = \int \frac{1}{x^2 \sqrt{x^2 - 9}} \, dx$

Step 1: Trigonometric Substitution

Since the integrand involves $\sqrt{x^2 - 9}$, which suggests a form similar to the Pythagorean identity, we can use a hyperbolic substitution (or a trigonometric substitution). The typical substitution for expressions of the form $\sqrt{x^2 - a^2}$ is:

$x = a \sec(\theta)$

In this case, we have $a^2 = 9$, so $a = 3$. Thus, the substitution is:

$x = 3 \sec(\theta)$

Step 2: Differentiating and Substituting

We now differentiate $x = 3 \sec(\theta)$ with respect to $\theta$:

$dx = 3 \sec(\theta) \tan(\theta) \, d\theta$

Substitute $x = 3 \sec(\theta)$ and $dx = 3 \sec(\theta) \tan(\theta) \, d\theta$ into the integral.

Also, substitute $\sqrt{x^2 - 9}$:

$\sqrt{x^2 - 9} = \sqrt{9 \sec^2(\theta) - 9} = \sqrt{9 (\sec^2(\theta) - 1)} = \sqrt{9 \tan^2(\theta)} = 3 \tan(\theta)$

Step 3: Substitute into the Integral

Now substitute all the terms into the original integral:

$I = \int \frac{1}{(3 \sec(\theta))^2 \cdot 3 \tan(\theta)} \cdot 3 \sec(\theta) \tan(\theta) \, d\theta$

Simplify the expression step-by-step:

$I = \int \frac{1}{9 \sec^2(\theta) \cdot 3 \tan(\theta)} \cdot 3 \sec(\theta) \tan(\theta) \, d\theta$

$I = \int \frac{1}{9 \sec(\theta) \cdot \tan(\theta)} \cdot \tan(\theta) \, d\theta$

The $\tan(\theta)$ terms cancel out:

$I = \int \frac{1}{9 \sec(\theta)} \, d\theta$

Now simplify the integrand:

$I = \frac{1}{9} \int \cos(\theta) \, d\theta$

Step 4: Integration

Now, integrate $\cos(\theta)$:

$I = \frac{1}{9} \sin(\theta) + C$

Step 5: Back-Substitute

We need to convert $\sin(\theta)$ back into terms of $x$. From the substitution $x = 3 \sec(\theta)$, we can use the identity $\sec^2(\theta) - \tan^2(\theta) = 1$, or simply recall that:

$\sin(\theta) = \frac{\sqrt{x^2 - 9}}{x}$

Thus, we can write:

$I = \frac{1}{9} \cdot \frac{\sqrt{x^2 - 9}}{x} + C$

Final Answer:

$I = \frac{\sqrt{x^2 - 9}}{9x} + C$

---

Would you like me to explain any part of the process in more detail? Here are a few related questions you may find helpful:

1. What other types of substitutions can be used to evaluate integrals involving $\sqrt{x^2 - a^2}$?
2. How do trigonometric identities play a role in simplifying integrals using substitution?
3. What is the significance of the hyperbolic identity $\sec^2(\theta) - 1 = \tan^2(\theta)$ in substitution problems?
4. How can you verify the correctness of your result by differentiating it?
5. How would the result change if the original integral had a different constant (for example, $x^2 - 16$ instead of $x^2 - 9$)?

Tip: For integrals involving $\sqrt{x^2 - a^2}$, always check for opportunities to use trigonometric or hyperbolic substitutions to simplify the expression before integrating!