The integral provided is:

$\int \left( 9x^2 - 2x - \frac{8}{x^9} + x^3 \right) dx$

Step-by-Step Solution:

We will integrate each term of the expression individually:

1. First Term: $9x^2$: $\int 9x^2 dx = \frac{9x^3}{3} = 3x^3$

2. Second Term: $-2x$: $\int -2x dx = \frac{-2x^2}{2} = -x^2$

3. Third Term: $-\frac{8}{x^9} = -8x^{-9}$: $\int -8x^{-9} dx = -8 \cdot \frac{x^{-8}}{-8} = x^{-8} = \frac{1}{x^8}$

4. Fourth Term: $x^3$: $\int x^3 dx = \frac{x^4}{4}$

Combine the Results:

Summing all terms together: $\int \left( 9x^2 - 2x - \frac{8}{x^9} + x^3 \right) dx = 3x^3 - x^2 + \frac{1}{x^8} + \frac{x^4}{4} + C$

where $C$ is the constant of integration.

Do you want further clarification or a graph of any part?

Here are 5 related questions for deeper exploration:

1. How would the result change if $x$ were constrained to positive values only?
2. Can this integral be verified using differentiation?
3. How does the presence of $x^{-9}$ influence the convergence of this integral on certain domains?
4. What are the practical applications of polynomial and rational integrals in physics?
5. Could substitution techniques be used to simplify any of these terms?

Tip: Always double-check your integration by differentiating the result—it should match the original function.