To solve the integral

$\int \frac{5}{2} x^{-\frac{3}{2}} \, dx,$

we follow these steps:

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Step 1: Rewrite the constant

The constant $\frac{5}{2}$ can be factored out of the integral:

$\frac{5}{2} \int x^{-\frac{3}{2}} \, dx.$

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Step 2: Apply the power rule of integration

The power rule for integration states: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for } n \neq -1.$

Here, $n = -\frac{3}{2}$. Adding 1 to the exponent:

$n + 1 = -\frac{3}{2} + 1 = -\frac{1}{2}.$

Now, the integral becomes: $\frac{5}{2} \cdot \frac{x^{-\frac{1}{2}}}{-\frac{1}{2}} + C.$

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Step 3: Simplify the expression

Dividing by $-\frac{1}{2}$ is the same as multiplying by $-2$:

$\frac{5}{2} \cdot (-2) x^{-\frac{1}{2}} + C = -5 x^{-\frac{1}{2}} + C.$

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Step 4: Express the result in terms of radicals

The exponent $-\frac{1}{2}$ corresponds to the reciprocal of the square root:

$x^{-\frac{1}{2}} = \frac{1}{\sqrt{x}}.$

Thus, the final answer is:

$\boxed{-\frac{5}{\sqrt{x}} + C}.$

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Would you like a detailed explanation of any step? Here are 5 questions to further expand this topic:

1. What is the power rule for integration and how is it applied?
2. Why can't the power rule be used for $n = -1$?
3. How do constants affect the integration process?
4. What are some real-world applications of power-law integrals?
5. How can you verify the result by differentiating the answer?

Tip: Always simplify the exponent to check if it represents a common radical (e.g., $-\frac{1}{2}$ for square root).