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We are given an integral to solve, and we need to find its general antiderivative. The integral is:

\[
\int \frac{x \sqrt{x} + \sqrt{x}}{x^2} \, dx
\]

### Step 1: Simplify the integrand

Let's first simplify the integrand:

\[
\frac{x \sqrt{x} + \sqrt{x}}{x^2} = \frac{x^{1.5} + x^{0.5}}{x^2}
\]

We can now break this expression into two parts:

\[
= \frac{x^{1.5}}{x^2} + \frac{x^{0.5}}{x^2}
\]

Simplifying both terms:

\[
= x^{1.5 - 2} + x^{0.5 - 2}
\]
\[
= x^{-0.5} + x^{-1.5}
\]

Thus, the integral becomes:

\[
\int (x^{-0.5} + x^{-1.5}) \, dx
\]

### Step 2: Integrate term by term

We'll now integrate each term separately:

1. \(\int x^{-0.5} \, dx\):  
   Using the power rule \(\int x^n \, dx = \frac{x^{n+1}}{n+1}\), we get:
   \[
   \int x^{-0.5} \, dx = \frac{x^{0.5}}{0.5} = 2 \sqrt{x}
   \]

2. \(\int x^{-1.5} \, dx\):  
   Again, using the power rule:
   \[
   \int x^{-1.5} \, dx = \frac{x^{-0.5}}{-0.5} = -2 x^{-0.5} = -\frac{2}{\sqrt{x}}
   \]

### Step 3: Write the general solution

The general antiderivative is:

\[
2 \sqrt{x} - \frac{2}{\sqrt{x}} + C
\]

### Step 4: Compare with the choices

Looking at the options provided, the correct answer is:

**Option D: \( 2\sqrt{x} - \frac{2}{\sqrt{x}} + C \)**

Would you like further clarification on any steps? Here are 5 related questions you could explore next:

1. How does the power rule for integration work?
2. What is the significance of the constant \(C\) in indefinite integrals?
3. How do we approach integrals with square roots?
4. Could we have used substitution or other techniques for this integral?
5. What are the common mistakes when simplifying integrals?

**Tip:** Always simplify the integrand as much as possible before integrating—it often makes the process much easier!

Find the most general antiderivative of \( \int \frac{x \sqrt{x} + \sqrt{x}}{x^2} dx \).

Learn how to find the general antiderivative of the integral \( \int \frac{x \sqrt{x} + \sqrt{x}}{x^2} dx \) using the power rule for integration and exponent simplification. This step-by-step solution explains the process and provides the correct answer.

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