To solve the integral \(\int \frac{1}{\sqrt{9 + 64x^2}} \, dx\), we can use a trigonometric substitution method.

### Steps to Solve:

1. **Identify the substitution**:
   The integral has the form \(\int \frac{1}{\sqrt{a^2 + b^2x^2}} \, dx\), which suggests the substitution \(x = \frac{a}{b} \tan(\theta)\), where \(a = 3\) and \(b = 8\).

   Set \(x = \frac{3}{8} \tan(\theta)\).

2. **Compute the differential**:
   \[
   dx = \frac{3}{8} \sec^2(\theta) \, d\theta
   \]

3. **Substitute into the integral**:
   Substitute \(x = \frac{3}{8} \tan(\theta)\) and \(dx = \frac{3}{8} \sec^2(\theta) \, d\theta\) into the integral:
   \[
   \int \frac{1}{\sqrt{9 + 64\left(\frac{3}{8} \tan(\theta)\right)^2}} \cdot \frac{3}{8} \sec^2(\theta) \, d\theta
   \]

   Simplify the expression inside the square root:
   \[
   \sqrt{9 + 64\left(\frac{9}{64} \tan^2(\theta)\right)} = \sqrt{9 + 9 \tan^2(\theta)} = 3\sqrt{1 + \tan^2(\theta)} = 3\sec(\theta)
   \]
   So the integral becomes:
   \[
   \int \frac{1}{3\sec(\theta)} \cdot \frac{3}{8} \sec^2(\theta) \, d\theta = \int \frac{1}{8} \sec(\theta) \, d\theta
   \]

4. **Integrate**:
   The integral of \(\sec(\theta)\) is known:
   \[
   \int \sec(\theta) \, d\theta = \ln|\sec(\theta) + \tan(\theta)| + C
   \]
   Thus, the integral becomes:
   \[
   \frac{1}{8} \ln|\sec(\theta) + \tan(\theta)| + C
   \]

5. **Substitute back**:
   Since \(x = \frac{3}{8} \tan(\theta)\), \(\tan(\theta) = \frac{8x}{3}\), and \(\sec(\theta) = \sqrt{1 + \tan^2(\theta)} = \sqrt{1 + \left(\frac{8x}{3}\right)^2} = \frac{\sqrt{9 + 64x^2}}{3}\), the final answer is:
   \[
   \frac{1}{8} \ln\left|\frac{\sqrt{9 + 64x^2}}{3} + \frac{8x}{3}\right| + C
   \]

This is the solution to the integral.

Would you like more details or have any questions about the steps? Here are some related questions you might find interesting:

1. How do we solve integrals using trigonometric substitution in general?
2. What is the general form of integrals involving \(\sqrt{a^2 + b^2x^2}\)?
3. What are the common trigonometric identities used in calculus?
4. Can you explain the integration of \(\sec(\theta)\) in more detail?
5. How do we handle integrals that don't easily fit common forms?

**Tip**: When dealing with square roots in integrals, look out for patterns that suggest trigonometric substitution. This can simplify the problem significantly.

Learn how to solve the integral of 1/sqrt(9 + 64x^2) step-by-step using trigonometric substitution. This problem covers integration techniques involving trigonometric functions and is suitable for advanced high school students or calculus students.

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