`int _(2) ^(3) x ^(4) dx =`

Options

A. `1/2`
B. `5/2`
C. `(5)/(211)`
D. `(211)/(5)`

Answer

To solve the integral \( \int_{2}^{3} x^{4} \, dx \), we will follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ I = \int_{2}^{3} x^{4} \, dx \]

### Step 2: Apply the power rule of integration
According to the power rule of integration, the integral of \( x^n \) is given by:
\[ \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \]
For our case, \( n = 4 \):
\[ \int x^{4} \, dx = \frac{x^{5}}{5} + C \]

### Step 3: Evaluate the definite integral
Now we will evaluate the definite integral from 2 to 3:
\[ I = \left[ \frac{x^{5}}{5} \right]_{2}^{3} \]
This means we will substitute the upper limit (3) and the lower limit (2) into the expression:
\[ I = \frac{3^{5}}{5} - \frac{2^{5}}{5} \]

### Step 4: Calculate \( 3^{5} \) and \( 2^{5} \)
Now we calculate \( 3^{5} \) and \( 2^{5} \):
\[ 3^{5} = 243 \quad \text{and} \quad 2^{5} = 32 \]

### Step 5: Substitute the values back into the integral
Substituting these values back into our expression for \( I \):
\[ I = \frac{243}{5} - \frac{32}{5} = \frac{243 - 32}{5} = \frac{211}{5} \]

### Final Answer
Thus, the value of the integral \( \int_{2}^{3} x^{4} \, dx \) is:
\[ \frac{211}{5} \]

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