Certainly! Let's evaluate the definite integral
[tex]\[
\int_1^2 \left( \frac{x^2}{3} + 7 \right) \, dx
\][/tex]
We will break this down into a few steps to find the exact value.
### Step 1: Split the Integral
First, we can split the integral into two separate integrals:
[tex]\[
\int_1^2 \left( \frac{x^2}{3} + 7 \right) dx = \int_1^2 \frac{x^2}{3} \, dx + \int_1^2 7 \, dx
\][/tex]
### Step 2: Evaluate Each Integral Separately
#### Integral 1: [tex]\(\int_1^2 \frac{x^2}{3} \, dx\)[/tex]
We can factor out [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\int_1^2 \frac{x^2}{3} \, dx = \frac{1}{3} \int_1^2 x^2 \, dx
\][/tex]
Now, we find the antiderivative of [tex]\(x^2\)[/tex], which is [tex]\(\frac{x^3}{3}\)[/tex]:
[tex]\[
\frac{1}{3} \left[ \frac{x^3}{3} \right]_1^2 = \frac{1}{3} \left( \frac{2^3}{3} - \frac{1^3}{3} \right) = \frac{1}{3} \left( \frac{8}{3} - \frac{1}{3} \right) = \frac{1}{3} \cdot \frac{7}{3} = \frac{7}{9}
\][/tex]
#### Integral 2: [tex]\(\int_1^2 7 \, dx\)[/tex]
For this integral, we simply integrate the constant 7:
[tex]\[
\int_1^2 7 \, dx = 7 \left[ x \right]_1^2 = 7 (2 - 1) = 7 (1) = 7
\][/tex]
### Step 3: Combine the Results
Finally, we add the results of the two integrals together:
[tex]\[
\frac{7}{9} + 7 = \frac{7}{9} + \frac{63}{9} = \frac{7 + 63}{9} = \frac{70}{9}
\][/tex]
Thus, the exact value of the definite integral
[tex]\[
\int_1^2 \left( \frac{x^2}{3} + 7 \right) dx
\][/tex]
is
[tex]\[
\boxed{\frac{70}{9}}
\][/tex]
