`int_(0)^(1)sqrt(x)dx=`

Options

A. 0
B. `(2)/(3)`
C. `(3)/(2)`
D. 1

Answer

To solve the definite integral \(\int_{0}^{1} \sqrt{x} \, dx\), we will follow these steps:

### Step 1: Rewrite the integrand
The integrand \(\sqrt{x}\) can be rewritten in terms of exponents:
\[ \sqrt{x} = x^{1/2} \]

### Step 2: Use the power rule for integration
We apply the power rule for integration, which states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
In our case, \(n = \frac{1}{2}\):
\[ \int x^{1/2} \, dx = \frac{x^{(1/2) + 1}}{(1/2) + 1} + C = \frac{x^{3/2}}{3/2} + C = \frac{2}{3} x^{3/2} + C \]

### Step 3: Evaluate the definite integral
Now we need to evaluate the definite integral from 0 to 1:
\[ \int_{0}^{1} \sqrt{x} \, dx = \left[ \frac{2}{3} x^{3/2} \right]_{0}^{1} \]

### Step 4: Substitute the limits
Now we substitute the upper and lower limits into the expression:
\[ = \frac{2}{3} (1)^{3/2} - \frac{2}{3} (0)^{3/2} \]
Calculating this gives:
\[ = \frac{2}{3} \cdot 1 - \frac{2}{3} \cdot 0 = \frac{2}{3} - 0 = \frac{2}{3} \]

### Final Answer
Thus, the value of the definite integral \(\int_{0}^{1} \sqrt{x} \, dx\) is:
\[ \frac{2}{3} \]

---