Evaluate `int _(1)^(3) x^2 dx`

Answer

To evaluate the integral \( \int_{1}^{3} x^2 \, dx \), we will follow these steps:

### Step 1: Identify the formula for integration
The formula for integrating \( x^n \) is given by:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
For our case, \( n = 2 \).

### Step 2: Apply the formula
Using the formula, we can write:
\[ \int x^2 \, dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3} \]

### Step 3: Set up the definite integral
Now, we need to evaluate this from the limits 1 to 3:
\[ \int_{1}^{3} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{1}^{3} \]

### Step 4: Evaluate at the upper limit
Substituting the upper limit \( x = 3 \):
\[ \frac{3^3}{3} = \frac{27}{3} = 9 \]

### Step 5: Evaluate at the lower limit
Substituting the lower limit \( x = 1 \):
\[ \frac{1^3}{3} = \frac{1}{3} \]

### Step 6: Subtract the lower limit from the upper limit
Now, we subtract the value at the lower limit from the value at the upper limit:
\[ 9 - \frac{1}{3} \]

### Step 7: Simplify the result
To perform the subtraction, we convert 9 into a fraction:
\[ 9 = \frac{27}{3} \]
So,
\[ \frac{27}{3} - \frac{1}{3} = \frac{27 - 1}{3} = \frac{26}{3} \]

### Final Answer
Thus, the value of the integral \( \int_{1}^{3} x^2 \, dx \) is:
\[ \frac{26}{3} \]

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