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!