$$ 3x^4-\frac{2}{x^2}+ $$$$ \sqrt{x}\differentialD x $$

It looks like you've written a mathematical expression that may be part of a larger equation or integral. Let's break it down:

The expression contains three distinct parts:

1. $ 3x^4 $: A polynomial term.
2. $ -\frac{2}{x^2} $: A rational term.
3. $ \sqrt{x} \differentialD x $: A square root term multiplied by a differential, which suggests it might be part of an integral.

If your intention was to integrate the entire expression, the corrected form would be:
$$ \int \left( 3x^4 - \frac{2}{x^2} + \sqrt{x} \right) \, dx $$

I will now compute the integral:

### Step-by-step integration:

1. **Integrating $ 3x^4 $:**
   \[
   \int 3x^4 \, dx = \frac{3}{5}x^5
   \]

2. **Integrating $ -\frac{2}{x^2} $:**
   This can be rewritten as $ -2x^{-2} $:
   \[
   \int -2x^{-2} \, dx = \frac{2}{x}
   \]

3. **Integrating $ \sqrt{x} $:**
   Recall that $ \sqrt{x} = x^{1/2} $:
   \[
   \int x^{1/2} \, dx = \frac{2}{3}x^{3/2}
   \]

### Final answer:
The integral is:
\[
\frac{3}{5}x^5 + \frac{2}{x} + \frac{2}{3}x^{3/2} + C
\]
where $ C $ is the constant of integration.

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Would you like more details on any part of this process?

### Related Questions:
1. How do you compute integrals involving logarithmic functions?
2. What are the steps to integrate a product of functions (e.g., using integration by parts)?
3. How do you handle definite integrals with limits of integration?
4. Can you explain the difference between improper integrals and regular integrals?
5. How do you calculate integrals of trigonometric functions?

### Tip:
When integrating, it's useful to rewrite terms in forms that are easier to handle, such as converting roots or fractions into exponent form.

Explore how to solve the integral of 3x^4 - \frac{2}{x^2} + \sqrt{x} \, dx. This step-by-step guide covers key calculus concepts, including polynomial integration, rational functions, and the square root function. Suitable for students in Grades 11-12 and college-level calculus.

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