To calculate the integral of \((3x + 5)^7\), we will use substitution. Here's the step-by-step process:

### Step 1: Substitution
Let:
\[
u = 3x + 5
\]
Now, differentiate \(u\) with respect to \(x\):
\[
\frac{du}{dx} = 3 \quad \text{or} \quad dx = \frac{du}{3}
\]

### Step 2: Rewrite the Integral
Substitute \(u = 3x + 5\) and \(dx = \frac{du}{3}\) into the integral:
\[
\int (3x + 5)^7 \, dx = \int u^7 \cdot \frac{du}{3}
\]
This simplifies to:
\[
\frac{1}{3} \int u^7 \, du
\]

### Step 3: Integrate
Now, integrate \(u^7\):
\[
\int u^7 \, du = \frac{u^8}{8}
\]
Thus, the integral becomes:
\[
\frac{1}{3} \cdot \frac{u^8}{8} = \frac{u^8}{24}
\]

### Step 4: Substitute \(u = 3x + 5\) back
Now, substitute \(u = 3x + 5\) back into the expression:
\[
\frac{(3x + 5)^8}{24} + C
\]
where \(C\) is the constant of integration.

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

Let me know if you'd like further clarification or details on any step!

Here are 5 related questions:
1. How do you find the derivative of \((3x + 5)^8\)?
2. Can you explain the method of substitution in integrals?
3. What is the integral of \((4x + 1)^5\)?
4. How do you evaluate definite integrals using substitution?
5. What happens if the exponent is negative in a similar integral?

**Tip:** When using substitution, always check if the variable you're substituting is linear; it makes the process simpler.

Learn how to calculate the integral of (3x + 5)^7 using the substitution method. This problem covers key calculus concepts like substitution in integration, and provides a step-by-step solution. Suitable for advanced high school and college students.

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