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Let's solve the given integral:

### Problem 29: \(\int (5x^2 + 1)(5x^3 + 3x - 8)^6 \, dx\)

This problem requires the method of substitution. The goal is to simplify the integral by substituting a part of the integrand.

1. **Substitution:**
   Let's choose \( u = 5x^3 + 3x - 8 \). Now differentiate \( u \) with respect to \( x \):
   \[
   \frac{du}{dx} = 15x^2 + 3 \quad \text{or} \quad du = (15x^2 + 3) \, dx
   \]
   Notice that \( 15x^2 + 3 \) can be factored as \( 3(5x^2 + 1) \). Thus,
   \[
   du = 3(5x^2 + 1) \, dx \quad \text{or} \quad \frac{du}{3} = (5x^2 + 1) \, dx
   \]

2. **Rewrite the integral:**
   Substitute \( u = 5x^3 + 3x - 8 \) into the integral:
   \[
   \int (5x^2 + 1)(5x^3 + 3x - 8)^6 \, dx = \int (5x^3 + 3x - 8)^6 \cdot \frac{du}{3}
   \]
   This simplifies to:
   \[
   \frac{1}{3} \int u^6 \, du
   \]

3. **Integrate:**
   Now integrate \( u^6 \):
   \[
   \int u^6 \, du = \frac{u^7}{7} + C
   \]
   So, the integral becomes:
   \[
   \frac{1}{3} \cdot \frac{u^7}{7} + C = \frac{1}{21} u^7 + C
   \]

4. **Substitute back:**
   Replace \( u \) with \( 5x^3 + 3x - 8 \):
   \[
   \int (5x^2 + 1)(5x^3 + 3x - 8)^6 \, dx = \frac{1}{21} (5x^3 + 3x - 8)^7 + C
   \]

### Final Answer:
\[
\int (5x^2 + 1)(5x^3 + 3x - 8)^6 \, dx = \frac{1}{21} (5x^3 + 3x - 8)^7 + C
\]

Would you like more details or have any other questions?

### Related Questions
1. How would you approach this integral if the exponent was different, such as \( (5x^3 + 3x - 8)^n \)?
2. Can this method be used for integrals involving trigonometric functions like \( \int (1 + \sin(x))(\cos(x))^5 \, dx \)?
3. What happens if the integrand includes a product of more than two polynomials?
4. How do you evaluate the integral if \( u \) substitution doesn't simplify the expression easily?
5. Can you use integration by parts instead of substitution for this problem?

### Tip:
When choosing a substitution, look for expressions inside the integrand whose derivative also appears elsewhere in the integrand. This simplifies the process significantly.

Explore the detailed step-by-step solution to evaluate the integral of (5x^2 + 1)(5x^3 + 3x - 8)^6 using the substitution method in integral calculus. This problem involves advanced techniques and is suitable for college-level mathematics.

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