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The problem requires evaluating an integral using both the given integration limits and the limits obtained by trigonometric substitution.

The integral given in the image is:
\[
\int \frac{\sqrt{x^2 - 36}}{x} \, dx
\]

### (a) The given integration limits:
The image shows the result as:
\[
\frac{\sqrt{3}}{2} \ln \left( \sqrt{3} + 2 \right)
\]

### (b) Trigonometric Substitution:
For the trigonometric substitution, we generally use:

1. Set \( x = 6 \sec \theta \), so \( dx = 6 \sec \theta \tan \theta \, d\theta \).
2. Then, \( x^2 - 36 = 36 (\sec^2 \theta - 1) = 36 \tan^2 \theta \), and \( \sqrt{x^2 - 36} = 6 \tan \theta \).

Substitute these into the integral:

\[
\int \frac{6 \tan \theta}{6 \sec \theta} \cdot 6 \sec \theta \tan \theta \, d\theta = \int 6 \tan^2 \theta \, d\theta
\]

Now, using the trigonometric identity \( \tan^2 \theta = \sec^2 \theta - 1 \):

\[
\int 6 (\sec^2 \theta - 1) \, d\theta = 6 \int (\sec^2 \theta - 1) \, d\theta
\]

This simplifies to:

\[
6 \left( \tan \theta - \theta \right) + C
\]

#### Limits for Trigonometric Substitution:

To find the new limits for \( \theta \) in terms of \( x \):

- When \( x = 6 \): \( 6 = 6 \sec \theta \implies \sec \theta = 1 \implies \theta = 0 \)
- When \( x = 12 \): \( 12 = 6 \sec \theta \implies \sec \theta = 2 \implies \theta = \frac{\pi}{3} \)

Thus, the limits in terms of \( \theta \) are from \( 0 \) to \( \frac{\pi}{3} \).

Would you like me to explain the next steps or provide further details?

### Additional Questions for Exploration:

1. What are the key trigonometric identities used in integral calculations?
2. How do we choose the appropriate trigonometric substitution for different integrals?
3. Why do trigonometric substitutions simplify integrals involving square roots?
4. How would the integral change if the expression under the square root was different (e.g., \( x^2 + 36 \) or \( 36 - x^2 \))?
5. What are other methods for solving integrals involving square roots besides trigonometric substitution?

### Tip:

When using trigonometric substitution, always sketch the corresponding right triangle to visualize the relationships between the trigonometric functions and the sides of the triangle. This helps in determining the correct substitutions and simplifying expressions.

Evaluate the integral using the given integration limits and the limits obtained by trigonometric substitution.

Learn how to evaluate integrals using trigonometric substitution and given integration limits. This example involves the integral of (sqrt(x^2 - 36) / x) with detailed steps, including substitution with x = 6 sec(θ) and limits transformation from x to θ. Suitable for students in Grades 11-12 and college-level calculus.

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