https://file.aiquickdraw.com/mathbot/file/1729937305066xfk8tq07.jpg

The image shows the integral:

\[
\int_8^{27} \frac{dx}{x - x^{1/3}} = \frac{3}{2} \ln \frac{8}{3}
\]

To solve this, let’s break down the approach:

### Step 1: Simplify the Integrand
Rewrite \( x - x^{1/3} \) by setting \( x = t^3 \), which implies \( dx = 3t^2 \, dt \). Then:
1. Substitute \( x = t^3 \) into the integral.
2. Solve and simplify the integral using the limits transformed by this substitution.
3. Confirm that it matches \( \frac{3}{2} \ln \frac{8}{3} \) as shown.

Would you like me to proceed with the full step-by-step solution?

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Here are some questions for further exploration:
1. What other substitution methods could simplify integrals of similar forms?
2. How does logarithmic integration apply to different function types?
3. What role does changing limits play in substitution integrals?
4. How can we solve improper integrals with similar structures?
5. What other techniques help in integrating functions involving roots?

**Tip:** For integrals involving complex expressions, substitution often simplifies the function by reducing variables or transforming the limits.

Evaluate the integral \( \int_8^{27} \frac{dx}{x - x^{1/3}} = \frac{3}{2} \ln \frac{8}{3} \)

Explore the detailed solution to evaluating the integral \( \int_8^{27} \frac{dx}{x - x^{1/3}} \) using the substitution method. This problem covers advanced integration techniques, including logarithmic integration and substitution methods. Suitable for undergraduate students.

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