Evaluate `int _(0)^(2) x^3 dx`

Answer

To evaluate the definite integral \( \int_{0}^{2} x^3 \, dx \), we will follow these steps:

### Step 1: Identify the integral
We need to evaluate the integral of \( x^3 \) from 0 to 2.

### Step 2: Use the power rule for integration
The power rule for integration states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( n \neq -1 \). In our case, \( n = 3 \).

### Step 3: Apply the power rule
Using the power rule, we find:
\[ \int x^3 \, dx = \frac{x^{3+1}}{3+1} = \frac{x^4}{4} + C \]

### Step 4: Evaluate the definite integral
Now we will evaluate the definite integral from 0 to 2:
\[ \int_{0}^{2} x^3 \, dx = \left[ \frac{x^4}{4} \right]_{0}^{2} \]

### Step 5: Substitute the limits
Substituting the upper limit (2) and the lower limit (0):
\[ = \frac{2^4}{4} - \frac{0^4}{4} \]

### Step 6: Calculate the values
Calculating \( 2^4 \):
\[ 2^4 = 16 \]
So,
\[ = \frac{16}{4} - 0 = 4 - 0 = 4 \]

### Final Answer
Thus, the value of the definite integral \( \int_{0}^{2} x^3 \, dx \) is:
\[ \boxed{4} \]