The problem asks to evaluate the integral:

$\int \frac{dx}{5x + 4}.$

Solution:

To solve this, we use the substitution method. Let:

$u = 5x + 4, \quad \text{so that} \quad du = 5dx, \quad \text{or} \quad dx = \frac{du}{5}.$

Substitute into the integral:

$\int \frac{dx}{5x + 4} = \int \frac{1}{u} \cdot \frac{du}{5} = \frac{1}{5} \int \frac{1}{u} \, du.$

The integral of $\frac{1}{u}$ is $\ln|u|$, so:

$\frac{1}{5} \int \frac{1}{u} \, du = \frac{1}{5} \ln|u| + C.$

Substitute back $u = 5x + 4$:

$\int \frac{dx}{5x + 4} = \frac{1}{5} \ln|5x + 4| + C.$

Final Answer:

$\boxed{\frac{1}{5} \ln|5x + 4| + C}$

Let me know if you'd like a more detailed explanation or have any questions!

Here are 5 related questions for deeper understanding:

1. How would you solve $\int \frac{dx}{ax + b}$ in general terms?
2. What happens if the integral were $\int \frac{1}{(5x+4)^2} \, dx$?
3. Can substitution always be used to solve rational function integrals like this?
4. How do absolute values in logarithms affect the integration result?
5. How would you evaluate a definite integral, e.g., $\int_0^1 \frac{dx}{5x + 4}$?

Tip: Always check if substitution simplifies an integral into a standard form—it’s one of the most powerful techniques!