We are tasked with solving the integral $\int (4x - 3)^4 \, dx$.

Steps for solving:

1. Substitution Method: Let $u = 4x - 3$, which means: $du = 4 \, dx \quad \text{or} \quad dx = \frac{du}{4}.$

2. Substitute: Substituting into the integral: $\int (4x - 3)^4 \, dx = \int u^4 \cdot \frac{du}{4}.$ Simplifying: $\frac{1}{4} \int u^4 \, du.$

3. Integrate: The integral of $u^4$ is: $\frac{u^5}{5}.$ Thus: $\frac{1}{4} \cdot \frac{u^5}{5} = \frac{u^5}{20}.$

4. Substitute back $u = 4x - 3$: $\frac{(4x - 3)^5}{20} + C.$

Thus, the correct answer is:

$\boxed{b. \, \frac{1}{20} (4x - 3)^5 + C}.$

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. How do you handle integrals of powers of linear functions?
2. What are other common integration techniques besides substitution?
3. How does the constant of integration work in indefinite integrals?
4. How would this problem change if the exponent were different?
5. Can you apply the same method to integrals involving trigonometric functions?

Tip: Always check if substitution can simplify an integral by reducing it to a basic form.