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To solve the problem of finding the anti-derivative of the function \( f(x) = \frac{6}{x^3} \), let's break it down.

### Step 1: Express the function in a simpler form
The given function \( f(x) = \frac{6}{x^3} \) can be rewritten as:
\[
f(x) = 6x^{-3}
\]

### Step 2: Find the anti-derivative (indefinite integral)
To find the anti-derivative, we need to integrate \( f(x) = 6x^{-3} \). Using the power rule for integration:
\[
\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad \text{(where \(n \neq -1\))}.
\]
For \( f(x) = 6x^{-3} \), apply the rule with \( n = -3 \):
\[
\int 6x^{-3} dx = 6 \cdot \frac{x^{-3+1}}{-3+1} + C = 6 \cdot \frac{x^{-2}}{-2} + C = -3x^{-2} + C
\]
Thus, the anti-derivative is:
\[
F(x) = -\frac{3}{x^2} + C
\]

### Step 3: Compare with the given options
Now, check the provided options:

1. \( F(x) = \frac{3}{2x^4} \)
2. \( F(x) = -\frac{3}{x^2} + 3 \) ← **This matches the form of our solution, where \( C = 3 \)**.
3. \( F(x) = 12x^{-2} + 1 \)
4. \( F(x) = 2x^4 \)

Clearly, the correct answer is:
\[
\boxed{F(x) = -\frac{3}{x^2} + 3}
\]

Would you like more details or have any further questions?

Here are some related questions to explore:

1. What is the power rule for derivatives and integrals?
2. How can constants of integration affect the anti-derivative?
3. What are common mistakes when finding anti-derivatives?
4. How do we integrate functions with fractional exponents?
5. How is the anti-derivative related to definite integrals?

**Tip:** When integrating, always remember to include the constant of integration, \( C \), unless dealing with a definite integral.

Which function is an anti-derivative of the function f(x) = 6/x^3?

This problem involves finding the anti-derivative of the function f(x) = 6/x^3. By applying the power rule for integration, the anti-derivative is calculated as F(x) = -3/x^2 + C. Check the options to find the correct solution.

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