To calculate the integral of \((3x + 5)^7\), we can use substitution.

### Steps:

1. **Substitute:**
   Let \( u = 3x + 5 \), so that \( \frac{du}{dx} = 3 \). This gives us:
   \[
   du = 3 \, dx \quad \text{or} \quad dx = \frac{du}{3}
   \]

2. **Rewrite the integral:**
   The integral becomes:
   \[
   \int (3x + 5)^7 \, dx = \int u^7 \cdot \frac{du}{3}
   \]
   \[
   = \frac{1}{3} \int u^7 \, du
   \]

3. **Integrate:**
   Now, integrate \( u^7 \):
   \[
   \frac{1}{3} \int u^7 \, du = \frac{1}{3} \cdot \frac{u^8}{8} = \frac{u^8}{24}
   \]

4. **Substitute \( u = 3x + 5 \) back:**
   \[
   \frac{(3x + 5)^8}{24} + C
   \]

### Final answer:
\[
\int (3x + 5)^7 \, dx = \frac{(3x + 5)^8}{24} + C
\]

Would you like further details or have any questions about this solution?

### Related Questions:
1. How do you integrate functions of the form \((ax + b)^n\)?
2. What is the substitution method in integration?
3. How does the chain rule apply to integration by substitution?
4. How do you find the derivative of \((3x + 5)^8\)?
5. What are other techniques to solve integrals when substitution doesn't work?

### Tip:
Always check your substitution carefully to ensure the differential terms match perfectly!

Learn how to calculate the integral of (3x + 5)^7 with a detailed step-by-step solution using the substitution method. This problem covers key calculus concepts, including integration and the power rule. Suitable for advanced high school students and first-year calculus students.

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