The integral `int_(1)^(5) x^(2)dx` is equal to

Options

A. `125/3`
B. `124/3`
C. `1/3`
D. `45`

Explanation

To solve the integral \( \int_{1}^{5} x^{2} \, dx \), we will follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ I = \int_{1}^{5} x^{2} \, dx \]

### Step 2: Apply the power rule for integration
Using the power rule for integration, which states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
for \( n = 2 \), we have:
\[ \int x^{2} \, dx = \frac{x^{3}}{3} + C \]

### Step 3: Evaluate the definite integral
Now we evaluate the definite integral from 1 to 5:
\[ I = \left[ \frac{x^{3}}{3} \right]_{1}^{5} \]
This means we will calculate:
\[ I = \frac{5^{3}}{3} - \frac{1^{3}}{3} \]

### Step 4: Calculate \( 5^{3} \) and \( 1^{3} \)
Calculating \( 5^{3} \):
\[ 5^{3} = 125 \]
Calculating \( 1^{3} \):
\[ 1^{3} = 1 \]

### Step 5: Substitute the values back into the equation
Substituting these values back into our equation gives:
\[ I = \frac{125}{3} - \frac{1}{3} \]

### Step 6: Combine the fractions
Now we combine the fractions:
\[ I = \frac{125 - 1}{3} = \frac{124}{3} \]

### Step 7: Final result
Thus, the value of the integral \( \int_{1}^{5} x^{2} \, dx \) is:
\[ \frac{124}{3} \]

### Summary
The integral \( \int_{1}^{5} x^{2} \, dx \) is equal to \( \frac{124}{3} \).

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