https://file.aiquickdraw.com/mathbot/file/17256709687133umulj99.jpg

The integral presented in the image is:

\[
\int_{4}^{8} \frac{x \, dx}{\sqrt{x^2 - 15}} = 6
\]

To solve this, we can use a substitution method. The hint provided suggests using the substitution \( u = \), which appears to be left incomplete in the image.

### Step-by-Step Solution:

1. **Substitution**:
   Let's substitute \( u = \sqrt{x^2 - 15} \). Then,
   \[
   u = \sqrt{x^2 - 15}
   \]
   Squaring both sides,
   \[
   u^2 = x^2 - 15 \quad \text{or} \quad x^2 = u^2 + 15.
   \]
   Differentiating with respect to \( x \),
   \[
   2x \, dx = 2u \, du \quad \text{or} \quad x \, dx = u \, du.
   \]

2. **Adjust Limits**:
   We also need to change the limits of integration according to the substitution \( u = \sqrt{x^2 - 15} \).

   - When \( x = 4 \), \( u = \sqrt{4^2 - 15} = \sqrt{16 - 15} = \sqrt{1} = 1 \).
   - When \( x = 8 \), \( u = \sqrt{8^2 - 15} = \sqrt{64 - 15} = \sqrt{49} = 7 \).

   Thus, the new limits of integration are from \( u = 1 \) to \( u = 7 \).

3. **Rewrite the Integral**:
   Substituting into the integral:
   \[
   \int_{1}^{7} \frac{u \, du}{u} = \int_{1}^{7} du = 7 - 1 = 6.
   \]

This confirms that the solution of the integral is indeed equal to 6, as stated.

### Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:

1. How does substitution simplify integrals with radicals?
2. What other substitution techniques can be used for integrals involving square roots?
3. Why does squaring the substitution help in dealing with radicals?
4. Can the same method be applied to integrals with different limits or more complex functions?
5. How does the change of limits affect the substitution process?

**Tip**: Always adjust your limits of integration when performing substitution, especially when dealing with definite integrals.

Solve the definite integral: \int_{4}^{8} \frac{x \, dx}{\sqrt{x^2 - 15}} = 6

Learn how to solve the definite integral \int_{4}^{8} \frac{x}{\sqrt{x^2 - 15}} dx using the substitution method. This calculus problem demonstrates integration techniques involving radicals, and is suitable for students in Grades 11-12 or early university-level calculus.

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