Math / Calculus

Question $\int_{2}^{4}\left(3 x^{2}+1\right) d x$ by integrating.

Studdy Solution

STEP 1

What is this asking? We need to find the definite integral of $3x^2 + 1$ from $x = 2$ to $x = 4$ . Watch out! Don't forget to apply the limits of integration after finding the indefinite integral!

STEP 2

1. Find the indefinite integral
2. Evaluate the definite integral

STEP 3

Let's break down our function into smaller, digestible pieces!
We have $3x^2 + 1$ .
We'll integrate each part separately.

STEP 4

The power rule for integration says: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ , where $C$ is the constant of integration.

STEP 5

Applying the power rule to $3x^2$ , we get: $\int 3x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} = 3 \cdot \frac{x^3}{3} = x^3$ .
Don't forget, we're keeping the constant of integration for the end!

STEP 6

The integral of $1$ with respect to $x$ is just $x$ .
Think of $1$ as $x^0$ , and applying the power rule gives us $\frac{x^{0+1}}{0+1} = x$ .

STEP 7

Putting it all together, the indefinite integral of $3x^2 + 1$ is $x^3 + x + C$ , where $C$ is our constant of integration.

STEP 8

Now for the exciting part: plugging in our limits of integration!
We have $\int_{2}^{4} (3x^2 + 1) \, dx = [x^3 + x]_{2}^{4}$ .

STEP 9

First, we substitute the upper limit, $x = 4$ : $(4)^3 + 4 = 64 + 4 = 68$ .

STEP 10

Next, we substitute the lower limit, $x = 2$ : $(2)^3 + 2 = 8 + 2 = 10$ .

STEP 11

Finally, we subtract the value at the lower limit from the value at the upper limit: $68 - 10 = 58$ .

STEP 12

Our final answer is $58$ !

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