Evaluate the following integrals :
`int x^(-3//2)dx`
(a and b are constant)

Answer

To evaluate the integral \( \int x^{-3/2} \, dx \), we will follow the standard procedure for integrating a power of \( x \).

### Step-by-Step Solution:

1. **Identify the integral**: We need to integrate the function \( x^{-3/2} \).

\[ \int x^{-3/2} \, dx \]

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

3. **Determine \( n + 1 \)**: Here, \( n = -\frac{3}{2} \). We need to calculate \( n + 1 \):

\[ n + 1 = -\frac{3}{2} + 1 = -\frac{3}{2} + \frac{2}{2} = -\frac{1}{2} \]

4. **Substitute into the power rule**: Now we can substitute \( n + 1 \) into the power rule:

\[ \int x^{-3/2} \, dx = \frac{x^{-1/2}}{-1/2} + C \]

5. **Simplify the expression**: The fraction simplifies as follows:

\[ \frac{x^{-1/2}}{-1/2} = -2x^{-1/2} \]

6. **Rewrite in a more recognizable form**: We can express \( x^{-1/2} \) as \( \frac{1}{\sqrt{x}} \):

\[ -2x^{-1/2} = -\frac{2}{\sqrt{x}} \]

7. **Final result**: Therefore, the integral evaluates to:

\[ \int x^{-3/2} \, dx = -\frac{2}{\sqrt{x}} + C \]

### Final Answer:
\[ \int x^{-3/2} \, dx = -\frac{2}{\sqrt{x}} + C \]