AP Calculus AB 6.9 Integration By 'u' Substitution (Example 9 e^x and Rationals)

### AP Calculus AB 6.9: Integration by "u" Substitution (Example 9: \(e^x\) and Rationals)

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#### Overview
In this example, students will learn how to apply the "u" substitution technique to integrate a function that involves the exponential function \(e^x\) combined with a rational expression. This illustrates how substitution can simplify complex integrals.

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#### Key Concepts

1. **Understanding "u" Substitution:**
- The "u" substitution technique is valuable for integrating functions that contain products of functions where one part can be differentiated easily.

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

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

4. **Substituting in Terms of \(u\):**
- Substitute \(u\) into the integral:
\[
\int e^x \cdot \frac{1}{(e^x + 1)^2} \, dx = \int e^x \cdot \frac{1}{u^2} \cdot \frac{du}{e^x}
\]
- Simplifying gives:
\[
\int \frac{1}{u^2} \, du
\]

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

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

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

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#### Conclusion
In this example, the integral of \(e^x \cdot \frac{1}{(e^x + 1)^2}\) illustrates the effectiveness of "u" substitution in simplifying the integration of functions involving exponentials and rational expressions. The steps demonstrate how an appropriate substitution can lead to a straightforward integral that can be evaluated easily, showcasing the utility of this technique in calculus.

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Nick Perich
Norristown Area High School
Norristown Area School District
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