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.