Let's start by simplifying the given expression:

The function to find the antiderivative of is: $\frac{9x^3 + 5x^5}{x^6}$

We can split this expression into two terms: $\frac{9x^3}{x^6} + \frac{5x^5}{x^6}$

Simplify each term:

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

So the expression becomes: $9x^{-3} + 5x^{-1}$

Now, we'll find the antiderivative of each term:

1. For $9x^{-3}$, the antiderivative is: $\frac{9x^{-3+1}}{-3+1} = \frac{9x^{-2}}{-2} = -\frac{9}{2}x^{-2}$

2. For $5x^{-1}$, the antiderivative is: $5 \ln|x|$

Thus, the overall antiderivative 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 5 related questions for further practice:

1. How would you find the antiderivative of $7x^{-4}$?
2. What is the antiderivative of $12x^{-1}$?
3. Can you explain why $5x^{-1}$ has a logarithmic antiderivative?
4. What is the antiderivative of $8x^{-2} + 3x^2$?
5. How do you compute the antiderivative of a sum of two functions?

Tip: When integrating powers of $x$, if the exponent is $-1$, the result involves a logarithmic function, $\ln|x|$, instead of following the usual power rule.