Evaluate : `int_(1)^(4) x^3 dx `

Options

A. `63.75`
B. `613.75`
C. `163.75`
D. `630.75`

Explanation

To evaluate the integral \( \int_{1}^{4} x^3 \, dx \), we will follow these steps:

### Step 1: Apply the power rule of integration
The power rule states that for any function \( x^n \), the integral is given by:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( C \) is the constant of integration. In our case, \( n = 3 \).

### Step 2: Calculate the integral
Using the power rule, we find:
\[ \int x^3 \, dx = \frac{x^{3+1}}{3+1} = \frac{x^4}{4} \]

### Step 3: Evaluate the definite integral from 1 to 4
Now we need to evaluate this from the limits 1 to 4:
\[ \int_{1}^{4} x^3 \, dx = \left[ \frac{x^4}{4} \right]_{1}^{4} \]

### Step 4: Substitute the upper limit
First, substitute the upper limit \( x = 4 \):
\[ \frac{4^4}{4} = \frac{256}{4} = 64 \]

### Step 5: Substitute the lower limit
Now, substitute the lower limit \( x = 1 \):
\[ \frac{1^4}{4} = \frac{1}{4} = 0.25 \]

### Step 6: Calculate the final result
Now, we subtract the value at the lower limit from the value at the upper limit:
\[ \int_{1}^{4} x^3 \, dx = 64 - 0.25 = 63.75 \]

Thus, the value of the integral \( \int_{1}^{4} x^3 \, dx \) is \( 63.75 \).

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