AP Calculus AB 6.9 Integration By 'u' Substitution (Example 10 Cubed Roots and Rationals)

### AP Calculus AB 6.9: Integration by "u" Substitution (Example 10: Cubed Roots and Rationals)

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#### Overview
In this example, students will apply the "u" substitution technique to integrate a function that involves a cubed root combined with a rational expression. This illustrates the method's flexibility in handling various types of integrals.

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#### Key Concepts

1. **Understanding "u" Substitution:**
- The "u" substitution technique is useful for simplifying integrals by transforming the integrand into a more manageable form.

2. **The Integral to Solve:**
- Consider the integral:
\[
\int \frac{x}{\sqrt[3]{x^2 + 1}} \, dx
\]

3. **Choosing the Substitution:**
- We choose:
\[
u = x^2 + 1 \quad \Rightarrow \quad du = 2x \, dx \quad \Rightarrow \quad dx = \frac{du}{2x}
\]

4. **Rewriting the Integral:**
- Substitute \(u\) into the integral:
\[
\int \frac{x}{\sqrt[3]{u}} \cdot \frac{du}{2x} = \int \frac{1}{2\sqrt[3]{u}} \, du
\]

5. **Integrating the Resulting Expression:**
- The integral \(\frac{1}{2\sqrt[3]{u}}\) can be rewritten using the power rule:
\[
\int \frac{1}{2} u^{-\frac{1}{3}} \, du = \frac{1}{2} \cdot \frac{u^{\frac{2}{3}}}{\frac{2}{3}} + C = \frac{3}{4} u^{\frac{2}{3}} + C
\]

6. **Back-Substituting for \(u\):**
- Replace \(u\) with the original expression:
\[
\frac{3}{4} (x^2 + 1)^{\frac{2}{3}} + C
\]

7. **Final Result:**
- Thus, the solution to the integral \(\int \frac{x}{\sqrt[3]{x^2 + 1}} \, dx\) is:
\[
\frac{3}{4} (x^2 + 1)^{\frac{2}{3}} + C
\]

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#### Conclusion
In this example, the integral of \(\frac{x}{\sqrt[3]{x^2 + 1}}\) demonstrates how "u" substitution can be effectively employed to simplify integrals involving cubed roots and rational functions. The process highlights the importance of identifying an appropriate substitution to facilitate easier integration, showcasing the versatility of this technique in calculus.

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Nick Perich
Norristown Area High School
Norristown Area School District
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