https://file.aiquickdraw.com/mathbot/file/1726280260160oojigtfn.jpg

The integral shown in the image is:

\[
\int \frac{dx}{x \sqrt{4x^2 - 25}}
\]

### Solution:

1. **Substitute**: 
   To simplify the square root term, let’s use the substitution:

   \[
   x = \frac{5}{2} \sec(\theta)
   \]
   This choice is motivated by the fact that \(\sec^2(\theta) - 1 = \tan^2(\theta)\), which will simplify the expression under the square root.

2. **Differentiate** \(x\):
   \[
   dx = \frac{5}{2} \sec(\theta) \tan(\theta) d\theta
   \]

3. **Substitute into the integral**:
   Now, we substitute \(x = \frac{5}{2} \sec(\theta)\) and \(dx = \frac{5}{2} \sec(\theta) \tan(\theta) d\theta\) into the integral.

   First, compute the term under the square root:

   \[
   4x^2 - 25 = 4 \left(\frac{5}{2} \sec(\theta)\right)^2 - 25 = 25 \sec^2(\theta) - 25 = 25 (\sec^2(\theta) - 1) = 25 \tan^2(\theta)
   \]

   Therefore, \(\sqrt{4x^2 - 25} = 5 \tan(\theta)\).

   Now the integral becomes:

   \[
   \int \frac{\frac{5}{2} \sec(\theta) \tan(\theta) d\theta}{\frac{5}{2} \sec(\theta) \cdot 5 \tan(\theta)} = \int \frac{d\theta}{5}
   \]

4. **Integrate**:
   \[
   \int \frac{d\theta}{5} = \frac{\theta}{5} + C
   \]

5. **Re-substitute** \(x = \frac{5}{2} \sec(\theta)\):
   From the substitution \(x = \frac{5}{2} \sec(\theta)\), we have \(\sec(\theta) = \frac{2x}{5}\). Therefore, \(\theta = \sec^{-1}\left(\frac{2x}{5}\right)\).

6. **Final answer**:
   The final result is:

   \[
   \frac{1}{5} \sec^{-1}\left(\frac{2x}{5}\right) + C
   \]

---

Would you like further details on any step or have questions about a specific part of the solution?

Here are 5 related questions to further expand your understanding:

1. How do trigonometric substitutions simplify integrals with square roots?
2. What is the general method for choosing a substitution in integrals involving \(\sqrt{a^2 - x^2}\), \(\sqrt{a^2 + x^2}\), or \(\sqrt{x^2 - a^2}\)?
3. What is the derivative of \(\sec^{-1}(x)\) and how is it derived?
4. Can you explain how the limits of integration would change in a definite integral using the same substitution?
5. How do hyperbolic trigonometric substitutions compare to standard trigonometric substitutions in integration?

**Tip**: When dealing with integrals involving square roots of quadratic expressions, using trigonometric substitutions based on common identities (\(\sec^2(\theta) - 1 = \tan^2(\theta)\)) often simplifies the problem dramatically.

Solve the integral ∫(dx / (x * sqrt(4x^2 - 25))) using trigonometric substitution with x = (5/2) sec(θ). This step-by-step solution covers key calculus concepts and theorems, making it ideal for undergraduate calculus students.

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