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We are given the following indefinite integral to evaluate:

\[
\int \left( \frac{9}{x^3} + \frac{4}{3x} \right) dx
\]

Let's break this down and solve step by step.

### Step 1: Split the integral
We can split the integral into two parts:

\[
\int \frac{9}{x^3} dx + \int \frac{4}{3x} dx
\]

### Step 2: Simplify each term

1. For \(\frac{9}{x^3}\), rewrite it as \(9x^{-3}\).
2. For \(\frac{4}{3x}\), it's already in a simple form, but we can factor out the constant \( \frac{4}{3} \).

Now we have:

\[
9 \int x^{-3} dx + \frac{4}{3} \int \frac{1}{x} dx
\]

### Step 3: Evaluate the integrals

1. The integral of \( x^{-3} \) is:

\[
\int x^{-3} dx = \frac{x^{-2}}{-2} = -\frac{1}{2x^2}
\]

So:

\[
9 \int x^{-3} dx = 9 \left( -\frac{1}{2x^2} \right) = -\frac{9}{2x^2}
\]

2. The integral of \( \frac{1}{x} \) is:

\[
\int \frac{1}{x} dx = \ln|x|
\]

So:

\[
\frac{4}{3} \int \frac{1}{x} dx = \frac{4}{3} \ln|x|
\]

### Step 4: Combine the results

Now, combining the two results, we get the final integral:

\[
-\frac{9}{2x^2} + \frac{4}{3} \ln|x| + C
\]

Where \(C\) is the constant of integration.

Would you like more details on any step? Here are some related questions to explore:

1. What is the rule for integrating powers of \(x\)?
2. How do you handle integrals involving logarithmic functions?
3. What are the general rules for integrating fractions?
4. How does the constant of integration work in indefinite integrals?
5. How would this problem change if it was a definite integral?

**Tip:** Always remember to include the absolute value inside logarithms when integrating \( \frac{1}{x} \), as the natural logarithm is only defined for positive values of \(x\).

Evaluate the indefinite integral ∫ (9/x^3 + 4/3x) dx

Learn how to evaluate the indefinite integral ∫ (9/x^3 + 4/3x) dx step by step. This problem involves the application of the power rule for integration and the logarithmic rule, suitable for advanced high school or undergraduate calculus students.

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