To solve the definite integral:

[\int_{2}^{4}(2x+3)dx]

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.

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

Learn how to solve the definite integral \(\int_{2}^{4}(2x + 3) \, dx\) with a step-by-step guide. This example covers key calculus concepts including antiderivatives and the Fundamental Theorem of Calculus. Suitable for students in Grades 10-12.

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