To find $F(x)$, the antiderivative of $f(x) = \frac{10}{x^3} - \frac{10}{x^6}$ with $F(1) = 0$, we will integrate $f(x)$ term by term.

Step 1: Rewrite $f(x)$ in a more convenient form

Rewrite each term with negative exponents: $f(x) = 10x^{-3} - 10x^{-6}$

Step 2: Integrate each term

We find the antiderivative $F(x)$ by integrating each part separately.

1. The integral of $10x^{-3}$ is: $\int 10x^{-3} \, dx = 10 \cdot \frac{x^{-2}}{-2} = -5x^{-2} = -\frac{5}{x^2}$

2. The integral of $-10x^{-6}$ is: $\int -10x^{-6} \, dx = -10 \cdot \frac{x^{-5}}{-5} = 2x^{-5} = \frac{2}{x^5}$

Thus, $F(x) = -\frac{5}{x^2} + \frac{2}{x^5} + C$

Step 3: Solve for $C$ using $F(1) = 0$

Substitute $x = 1$ and $F(1) = 0$: $F(1) = -\frac{5}{1^2} + \frac{2}{1^5} + C = -5 + 2 + C = 0$ $C = 3$

Final Answer

The function $F(x)$ is: $F(x) = -\frac{5}{x^2} + \frac{2}{x^5} + 3$

Would you like more details on any step, or do you have further questions?

Here are five related questions that could further expand your understanding:

1. How would you verify that $F(x)$ is indeed the antiderivative of $f(x)$?
2. What would change if the initial condition was $F(1) = 1$ instead of $F(1) = 0$?
3. How does the integration process change if $f(x)$ has additional terms with different exponents?
4. What is the significance of the constant $C$ in indefinite integrals?
5. How does the antiderivative behave as $x \to \infty$?

Tip: When dealing with powers of $x$ in integration, rewrite terms with negative exponents to simplify the integration process.