The anti-derivative of `(sqrt(x) + (1)/(sqrt(x)))` equals

Options

A. `(1)/(3)x^(1//3) + 2x^(1//2) + C`
B. `(2)/(3)x^(2//3) + (1)/(2)x^(2) + C`
C. `(2)/(3)x^(3//2) + 2x^(1//2) + C`
D. `(3)/(2)x^(3//2) + (1)/(2)x^(1//2) + C`

Answer

To find the anti-derivative (or integral) of the function \( \sqrt{x} + \frac{1}{\sqrt{x}} \), we will break it down into simpler parts and integrate each part separately.

### Step-by-step Solution:

1. **Rewrite the expression**:
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = x^{1/2} + x^{-1/2} \]

2. **Set up the integral**:
\[ \int \left( x^{1/2} + x^{-1/2} \right) \, dx \]

3. **Integrate each term separately**:
- For the first term \( x^{1/2} \):
\[ \int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \]
- For the second term \( x^{-1/2} \):
\[ \int x^{-1/2} \, dx = \frac{x^{-1/2 + 1}}{-1/2 + 1} = \frac{x^{1/2}}{1/2} = 2 x^{1/2} \]

4. **Combine the results**:
\[ \int \left( x^{1/2} + x^{-1/2} \right) \, dx = \frac{2}{3} x^{3/2} + 2 x^{1/2} + C \]
where \( C \) is the constant of integration.

5. **Final result**:
\[ \int \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right) \, dx = \frac{2}{3} x^{3/2} + 2 \sqrt{x} + C \]