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The given integral is:

\[
\int_4^9 \frac{x^{\frac{1}{4}}}{(13 - x)^{\frac{1}{4}} + x^{\frac{1}{4}}} \, dx
\]

We can start by analyzing the structure of the integrand and look for potential simplifications or substitutions to solve it. Let's proceed with finding a solution.The integral does not simplify easily using elementary functions, meaning it is complex and may require numerical methods for an explicit evaluation.

We can proceed by approximating the value of this integral numerically. Let me calculate that next.The numerical value of the integral is approximately:

\[
\int_4^9 \frac{x^{\frac{1}{4}}}{(13 - x)^{\frac{1}{4}} + x^{\frac{1}{4}}} \, dx \approx 2.5
\]

Would you like further clarification or details on any part of the solution?

Here are 5 questions to deepen your understanding:
1. What is the importance of substitution in solving integrals?
2. How do numerical methods help when solving complex integrals?
3. What is the role of the limits of integration in definite integrals?
4. Can you explain the significance of the fractional powers in this integral?
5. How would you approach integrals that involve irrational functions?

**Tip:** When an integral cannot be solved symbolically, numerical integration (like Simpson's rule or the trapezoidal rule) is often a powerful alternative.

Integrate the function x^(1/4) / [(13 - x)^(1/4) + x^(1/4)] from 4 to 9.

Explore the solution to the definite integral of x^(1/4) / [(13 - x)^(1/4) + x^(1/4)] from 4 to 9. Understand the use of fractional exponents and numerical methods to approximate the integral's value, suitable for college-level calculus students.

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