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The integral to solve for question 29 is:  

\[
\int_{-6}^{4} x \, dx
\]

### Solution:  
The integral of \(x\) with respect to \(x\) is given by:  
\[
\int x \, dx = \frac{x^2}{2} + C
\]

Using the Fundamental Theorem of Calculus to evaluate the definite integral:  
\[
\int_{-6}^{4} x \, dx = \left[\frac{x^2}{2}\right]_{-6}^{4}
\]

1. First, evaluate at the upper limit (\(x = 4\)):  
\[
\frac{4^2}{2} = \frac{16}{2} = 8
\]

2. Next, evaluate at the lower limit (\(x = -6\)):  
\[
\frac{(-6)^2}{2} = \frac{36}{2} = 18
\]

3. Subtract the value at the lower limit from the value at the upper limit:  
\[
\int_{-6}^{4} x \, dx = 8 - 18 = -10
\]

### Final Answer:  
\[
\int_{-6}^{4} x \, dx = -10
\]

Let me know if you want further clarification or details!  

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### Related Questions:
1. How do we interpret the result of a definite integral geometrically?  
2. How would the result change if the bounds were swapped (i.e., \(\int_{4}^{-6} x \, dx\))?  
3. What is the significance of a negative value in a definite integral?  
4. How does the presence of constants (e.g., \(\int_{-6}^{4} (x+c) \, dx\)) affect the integral?  
5. Can this integral be split into multiple integrals for easier computation?  

### Tip:  
When solving definite integrals, always simplify the function first (if possible) and double-check the calculation of limits to avoid sign errors.

Learn how to evaluate the definite integral \(\int_{-6}^{4} x \, dx\) using the Fundamental Theorem of Calculus. This step-by-step solution demonstrates key integration concepts for high school students in Grades 11-12.

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