how he went from: *x*3+*x*−*x to x*−x / 1+*x*2*x*​

Let's break down the step from the integrand \(\frac{x^3 + x - x}{1 + x^2}\) to \(\int \left( x - \frac{x}{1 + x^2} \right) \, dx\):

1. **Original integrand:**
   \[
   \int \frac{x^3 + x - x}{1 + x^2} \, dx
   \]

2. **Combine like terms in the numerator:**
   \[
   x^3 + x - x = x^3
   \]

3. **Simplify the integrand:**
   \[
   \frac{x^3 + x - x}{1 + x^2} = \frac{x^3}{1 + x^2}
   \]

4. **Split the fraction:**
   \[
   \frac{x^3}{1 + x^2} = \frac{x \cdot x^2}{1 + x^2} = \frac{x \cdot (1 + x^2 - 1)}{1 + x^2}
   \]

5. **Separate the terms:**
   \[
   \frac{x \cdot (1 + x^2 - 1)}{1 + x^2} = \frac{x \cdot (1 + x^2)}{1 + x^2} - \frac{x}{1 + x^2}
   \]

6. **Simplify the first term:**
   \[
   \frac{x \cdot (1 + x^2)}{1 + x^2} = x
   \]

7. **Rewrite the integrand:**
   \[
   \frac{x^3}{1 + x^2} = x - \frac{x}{1 + x^2}
   \]

Therefore, we have:
\[
\int \frac{x^3 + x - x}{1 + x^2} \, dx = \int \left( x - \frac{x}{1 + x^2} \right) \, dx
\]

This shows how the integrand \(\frac{x^3 + x - x}{1 + x^2}\) was transformed to \(x - \frac{x}{1 + x^2}\).

Do you want more details or have any other questions? Here are some questions you might ask:

1. Can you show the detailed steps for another integral transformation?
2. What are some common techniques for simplifying integrands?
3. How does splitting fractions help in integration?
4. Are there any general rules for recognizing how to break down complex integrands?
5. How can we apply this method to other types of integrals?

**Tip:** When simplifying complex integrands, always look for opportunities to separate or combine terms in the numerator and denominator. This often reveals simpler forms that are easier to integrate.

Learn how to transform the integral from (x^3 + x - x) / (1 + x^2) to x - (x / (1 + x^2)). This detailed step-by-step solution covers integration techniques, including simplification and fraction decomposition. Suitable for college-level students studying calculus.

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