Let's simplify the expression before integrating:

Given:
$\int \left(\frac{x^3}{3} - x\right) \cdot \frac{1}{x} \, dx$

Step 1: Simplify the integrand

Distribute $\frac{1}{x}$ to each term inside the parentheses:

$\int \left( \frac{x^3}{3x} - \frac{x}{x} \right) \, dx$

This simplifies to:

$\int \left( \frac{x^2}{3} - 1 \right) \, dx$

Step 2: Integrate term by term

$\int \frac{x^2}{3} \, dx - \int 1 \, dx$

Now integrate each part:

• For $\frac{x^2}{3}$, the integral is: $\frac{1}{3} \cdot \frac{x^3}{3} = \frac{x^3}{9}$
• For $1$, the integral is: $x$

Step 3: Combine the results

$\frac{x^3}{9} - x + C$

where $C$ is the constant of integration.

Final Answer

$\frac{x^3}{9} - x + C$

Would you like further explanation or have any other questions?

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Here are 5 related questions to deepen your understanding:

1. How do you apply the power rule for integration?
2. What are the steps to simplify complex integrands before integrating?
3. How do you handle definite integrals with similar expressions?
4. What is the role of the constant of integration in indefinite integrals?
5. How does changing variables affect integration problems?

Tip: Always simplify the integrand if possible—it makes the integration process much smoother.