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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.

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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.

Find an antiderivative of (9x^3 + 5x^5) / x^6 in the variable x where x ≠ 0.

Learn how to find the antiderivative of (9x^3 + 5x^5) / x^6 in the variable x by simplifying the expression and applying integration rules. This detailed solution walks through the steps of solving using the power rule and logarithmic integration.

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