Evaluate the following :
`int_(0)^(oo)x^(-1//2)dx`

Answer

To evaluate the integral \( I = \int_{0}^{\infty} x^{-\frac{1}{2}} \, dx \), we will follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ I = \int_{0}^{\infty} x^{-\frac{1}{2}} \, dx \]

### Step 2: Find the antiderivative
The integral of \( x^n \) is given by:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(for } n \neq -1\text{)} \]
In our case, \( n = -\frac{1}{2} \). Therefore, we compute:
\[ \int x^{-\frac{1}{2}} \, dx = \frac{x^{-\frac{1}{2} + 1}}{-\frac{1}{2} + 1} = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2x^{\frac{1}{2}} + C \]

### Step 3: Evaluate the definite integral
Now we evaluate the definite integral from \( 0 \) to \( \infty \):
\[ I = \left[ 2x^{\frac{1}{2}} \right]_{0}^{\infty} \]

### Step 4: Calculate the limits
1. Evaluate at the upper limit \( x = \infty \):
\[ 2(\infty)^{\frac{1}{2}} = 2 \cdot \infty = \infty \]
2. Evaluate at the lower limit \( x = 0 \):
\[ 2(0)^{\frac{1}{2}} = 2 \cdot 0 = 0 \]

### Step 5: Combine the results
Now, we combine the results from the limits:
\[ I = \infty - 0 = \infty \]

### Final Result
Thus, the value of the integral is:
\[ \int_{0}^{\infty} x^{-\frac{1}{2}} \, dx = \infty \]