To find the antiderivative of the expression

$\frac{9x^{3} + 5x^{5}}{x^6}$

we first simplify it by dividing each term in the numerator by $x^6$:

$\frac{9x^{3}}{x^6} + \frac{5x^{5}}{x^6} = 9x^{-3} + 5x^{-1}$

Now we find the antiderivative of each term separately with respect to $x$:

1. For $9x^{-3}$: $\int 9x^{-3} \, dx = 9 \cdot \frac{x^{-2}}{-2} = -\frac{9}{2}x^{-2}$

2. For $5x^{-1}$: $\int 5x^{-1} \, dx = 5 \ln |x|$

Therefore, the antiderivative of the expression is:

$-\frac{9}{2}x^{-2} + 5 \ln |x| + C$

where $C$ is the constant of integration.

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:

1. How would the approach change if we were finding a definite integral over a specific interval?
2. What are the rules for integrating functions with negative exponents?
3. How would we handle integration if additional terms were added to the numerator?
4. Why do we include an absolute value around $x$ when integrating $x^{-1}$?
5. How can we verify that this antiderivative is correct by differentiation?

Tip: Always simplify expressions as much as possible before integrating to avoid complex calculations.