To solve the definite integral $\int_{2}^{4}(2x + 3) \, dx$, follow these steps:

Step 1: Find the antiderivative

The function to integrate is $2x + 3$. The antiderivative (or indefinite integral) of this function can be found as follows:

$\int (2x + 3) \, dx = \int 2x \, dx + \int 3 \, dx$

Now, integrate each term:

$\int 2x \, dx = x^2$

$\int 3 \, dx = 3x$

So, the antiderivative of $2x + 3$ is:

$x^2 + 3x + C$

Step 2: Evaluate the definite integral

To evaluate the definite integral from 2 to 4, use the antiderivative:

$\int_{2}^{4}(2x + 3) \, dx = \left[ x^2 + 3x \right]_{2}^{4}$

This means you substitute the upper limit $4$ and the lower limit $2$ into the antiderivative and then subtract:

[ \left[ (4)^2 + 3(4) \right] - \left[ (2)^2 + 3(2) \right] ]

Step 3: Compute the values

Calculate each expression:

For $x = 4$:

$(4)^2 + 3(4) = 16 + 12 = 28$

For $x = 2$:

$(2)^2 + 3(2) = 4 + 6 = 10$

Step 4: Subtract the lower limit result from the upper limit result

$28 - 10 = 18$

So, the value of the definite integral is:

$\int_{2}^{4}(2x + 3) \, dx = 18$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you find the antiderivative of a more complex polynomial function?
2. Can you apply integration by parts to solve a definite integral?
3. What are the properties of definite integrals that simplify computations?
4. How can definite integrals be used to calculate the area under a curve?
5. What is the Fundamental Theorem of Calculus, and how does it relate to definite integrals?

Tip: When solving definite integrals, always ensure to evaluate the antiderivative correctly at both the upper and lower limits to avoid simple errors.