https://file.aiquickdraw.com/mathbot/file/1726285037051quaxh5l9.jpg

The expression in the image represents the integral:

\[
\frac{1}{5} \left( \int_1^5 \frac{\sqrt{x^2 - 1}}{x} \, dx \right)
\]

### Steps to solve:

1. **Simplifying the integrand**:
   The integral is:
   \[
   \int_1^5 \frac{\sqrt{x^2 - 1}}{x} \, dx
   \]
   This is a standard integral that can be solved using the substitution \( x = \sec(\theta) \). This will simplify the integrand.

2. **Substitution**:
   Let \( x = \sec(\theta) \), so that \( dx = \sec(\theta) \tan(\theta) d\theta \) and \( \sqrt{x^2 - 1} = \tan(\theta) \).
   The limits of integration will change accordingly:
   - When \( x = 1 \), \( \theta = 0 \).
   - When \( x = 5 \), \( \theta = \sec^{-1}(5) \).

3. **Solving the integral**:
   After substitution, the integral becomes:
   \[
   \int_0^{\sec^{-1}(5)} \tan^2(\theta) \, d\theta
   \]
   Using the identity \( \tan^2(\theta) = \sec^2(\theta) - 1 \), the integral can be split and solved.

4. **Final expression**:
   After solving the integral and evaluating the bounds, multiply the result by \( \frac{1}{5} \) to complete the calculation.

Would you like a detailed step-by-step solution for the substitution and final evaluation?

### Related Questions:
1. What is the technique used for integrals involving \( \sqrt{x^2 - a^2} \)?
2. How do trigonometric substitutions simplify integrals with square roots?
3. What are the general rules for changing limits of integration during substitution?
4. How is the identity \( \tan^2(\theta) = \sec^2(\theta) - 1 \) used in integration?
5. How do you compute definite integrals involving inverse trigonometric functions?

**Tip:** Substitutions using trigonometric identities are highly useful for integrals involving square roots of quadratic expressions.

Evaluate the integral \( \frac{1}{5} \left( \int_1^5 \frac{\sqrt{x^2 - 1}}{x} dx \right) \).

This guide explains how to solve the integral \( \frac{1}{5} \left( \int_1^5 \frac{\sqrt{x^2 - 1}}{x} dx \right) \) using trigonometric substitution. Learn about the key concepts, including trigonometric identities and solving definite integrals involving square roots of quadratics. Suitable for college-level calculus students.

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