To evaluate the definite integral of $2x + 1$ from 0 to 4, we follow these steps:

$\int_0^4 (2x + 1) \, dx$

Step 1: Find the indefinite integral

We first find the indefinite integral of $2x + 1$.

$\int (2x + 1) \, dx = x^2 + x + C$

Step 2: Apply the limits of integration

Now, we evaluate this result between the limits 0 and 4:

$\left[ x^2 + x \right]_0^4 = \left( 4^2 + 4 \right) - \left( 0^2 + 0 \right)$

$= (16 + 4) - (0 + 0) = 20$

Final Answer:

The value of the definite integral is $\boxed{20}$.

Would you like further details or explanations on any part? Here are five related questions to deepen your understanding:

1. How do you find the indefinite integral of a polynomial function?
2. How does the Fundamental Theorem of Calculus apply to this integral?
3. What are the geometrical interpretations of definite integrals?
4. How do we approach integrating more complex functions?
5. What is the difference between indefinite and definite integrals?

Tip: Always verify the bounds of integration when solving definite integrals to avoid simple errors!