The problem is to evaluate the integral:

$\int 5^{4x} \, dx$

Step-by-step solution:

1. Rewrite the integrand: The base $5^{4x}$ can be rewritten using properties of exponents: $5^{4x} = \left(5^x\right)^4$

   However, it is simpler to directly work with $5^{4x}$ in its original form.

2. Recall the general formula for exponential integrals: If you are integrating $a^{f(x)} \cdot f'(x)$, the result is: $\int a^{f(x)} \, dx = \frac{a^{f(x)}}{\ln(a)} \cdot \frac{1}{f'(x)}$ where $a > 0$ and $a \neq 1$.

3. Differentiate $4x$: Here, $f(x) = 4x$, so $f'(x) = 4$.

4. Set up the integral: Using the formula above for $a = 5$, $f(x) = 4x$, and $f'(x) = 4$: $\int 5^{4x} \, dx = \frac{5^{4x}}{\ln(5)} \cdot \frac{1}{4}$

5. Simplify: Combine terms to get the final answer: $\int 5^{4x} \, dx = \frac{5^{4x}}{4 \ln(5)} + C$ where $C$ is the constant of integration.

Final Answer:

$\boxed{\int 5^{4x} \, dx = \frac{5^{4x}}{4 \ln(5)} + C}$

Would you like further details or clarifications? Here are some related questions for practice:

1. What is the integral of $5^{2x}$?
2. How do you integrate $e^{4x}$?
3. Can you derive the formula for integrating $a^{f(x)}$?
4. What happens if the base is $e$ instead of $5$?
5. How do logarithms affect the integration of