`int (1)/(sqrt(x)) dx`

Answer

To solve the integral \(\int \frac{1}{\sqrt{x}} \, dx\), follow these steps:

### Step-by-Step Solution:

1. **Rewrite the integrand:**
\[ \int \frac{1}{\sqrt{x}} \, dx = \int x^{-\frac{1}{2}} \, dx \]

2. **Apply the power rule for integration:**
The power rule for integration states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n \neq -1\).

Here, \(n = -\frac{1}{2}\).

3. **Add 1 to the exponent:**
\[ x^{-\frac{1}{2} + 1} = x^{\frac{1}{2}} \]

4. **Divide by the new exponent:**
\[ \int x^{-\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C \]

5. **Simplify the fraction:**
\[ \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2x^{\frac{1}{2}} \]

6. **Rewrite the exponent as a square root:**
\[ 2x^{\frac{1}{2}} = 2\sqrt{x} \]

7. **Include the constant of integration:**
\[ \int \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x} + C \]

### Final Answer:
\[ \int \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x} + C \]