`int_(1)^(4)xsqrt(x)dx=?`

Options

A. `12.8`
B. `12.4`
C. `7`
D. none of these

Explanation

To solve the integral \( \int_{1}^{4} x \sqrt{x} \, dx \), we can follow these steps:

### Step 1: Rewrite the integrand
The integrand \( x \sqrt{x} \) can be rewritten using exponents. We know that \( \sqrt{x} = x^{1/2} \). Therefore, we can express the integrand as:
\[ x \sqrt{x} = x \cdot x^{1/2} = x^{1 + 1/2} = x^{3/2} \]

### Step 2: Set up the integral
Now, we can set up the integral with the new expression:
\[ \int_{1}^{4} x \sqrt{x} \, dx = \int_{1}^{4} x^{3/2} \, dx \]

### Step 3: Apply the power rule for integration
To integrate \( x^{3/2} \), we use the power rule for integration, which states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
For our case, \( n = \frac{3}{2} \):
\[ \int x^{3/2} \, dx = \frac{x^{3/2 + 1}}{3/2 + 1} = \frac{x^{5/2}}{5/2} = \frac{2}{5} x^{5/2} \]

### Step 4: Evaluate the definite integral
Now we need to evaluate the definite integral from 1 to 4:
\[ \int_{1}^{4} x^{3/2} \, dx = \left[ \frac{2}{5} x^{5/2} \right]_{1}^{4} \]
Calculating the upper limit:
\[ \frac{2}{5} (4^{5/2}) = \frac{2}{5} (32) = \frac{64}{5} \]
Calculating the lower limit:
\[ \frac{2}{5} (1^{5/2}) = \frac{2}{5} (1) = \frac{2}{5} \]

### Step 5: Subtract the lower limit from the upper limit
Now we subtract the lower limit from the upper limit:
\[ \int_{1}^{4} x^{3/2} \, dx = \frac{64}{5} - \frac{2}{5} = \frac{64 - 2}{5} = \frac{62}{5} \]

### Final Answer
Thus, the value of the integral \( \int_{1}^{4} x \sqrt{x} \, dx \) is:
\[ \frac{62}{5} \]