integrate 1/(6root(x-6)) lower bound of 6, upper bound 8

To compute the definite integral of \( \frac{1}{6\sqrt{x-6}} \) from 6 to 8, we can follow these steps:

1. **Rewrite the integral:**

   The given integral is:
   \[
   \int_{6}^{8} \frac{1}{6\sqrt{x-6}} \, dx
   \]
   Let’s perform a substitution to simplify this expression.

2. **Substitute:**
   
   Let \( u = x - 6 \), then \( du = dx \).
   When \( x = 6 \), \( u = 0 \), and when \( x = 8 \), \( u = 2 \).
   
   The integral becomes:
   \[
   \int_{0}^{2} \frac{1}{6\sqrt{u}} \, du
   \]

3. **Solve the integral:**

   Now, we can integrate \( \frac{1}{6\sqrt{u}} \), which is equivalent to:
   \[
   \frac{1}{6} \int_{0}^{2} u^{-\frac{1}{2}} \, du
   \]
   The integral of \( u^{-\frac{1}{2}} \) is \( 2u^{\frac{1}{2}} \). So we get:
   \[
   \frac{1}{6} \cdot 2u^{\frac{1}{2}} = \frac{1}{3} \sqrt{u}
   \]

4. **Evaluate the definite integral:**

   Now, we evaluate \( \frac{1}{3} \sqrt{u} \) from 0 to 2:
   \[
   \left[ \frac{1}{3} \sqrt{u} \right]_{0}^{2} = \frac{1}{3} \left( \sqrt{2} - \sqrt{0} \right) = \frac{\sqrt{2}}{3}
   \]

5. **Conclusion:**

   The value of the definite integral is:
   \[
   \frac{\sqrt{2}}{3}
   \]

Would you like further explanation on any step?

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Here are 5 related questions to expand your understanding:

1. How would you approach solving an integral with a different square root term, such as \( \frac{1}{\sqrt{x+3}} \)?
2. What changes if the upper bound of the integral was 10 instead of 8?
3. Can you solve an indefinite integral of \( \frac{1}{6\sqrt{x-6}} \)?
4. What is the physical meaning of integrals in the context of area under a curve?
5. How does substitution help simplify more complex integrals?

**Tip:** When dealing with square roots in integrals, substitution often makes the process much easier by turning them into simple powers.

This problem demonstrates how to compute the definite integral of 1/(6√(x-6)) from 6 to 8 using the substitution method. It includes detailed steps, substitution of u = x - 6, and evaluation of the result, suitable for undergraduate-level calculus students.

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