AP Calculus AB TOPIC 6.9 Integrating Using Substitution

### Learning Objective: FUN-6.D

#### Overview
Learning Objective FUN-6.D focuses on using substitution techniques to determine indefinite integrals and evaluate definite integrals. This objective emphasizes the importance of recognizing when to apply substitution or rearrangement strategies to simplify integrands.

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#### Key Concepts

1. **Indefinite Integrals:**
- Students should be able to determine indefinite integrals for functions requiring substitution or rearrangement into equivalent forms.
- Substitution is a fundamental technique used to find antiderivatives by replacing a complicated expression with a simpler variable.

2. **Definite Integrals:**
- Students should understand that when evaluating definite integrals, they must make corresponding changes to the limits of integration based on the substitution applied.
- This ensures the correct calculation of the area under the curve represented by the integral.

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#### Essential Knowledge

1. **Substitution of Variables:**
- FUN-6.D.1: This technique involves choosing a new variable to simplify the integrand. It helps in transforming a complex function into a more manageable one, making integration straightforward.
- For example, if \( u = g(x) \), then the integral can be expressed in terms of \( u \), allowing for easier evaluation.

2. **Adjusting Limits of Integration:**
- FUN-6.D.2: When using substitution for definite integrals, the limits of integration must be adjusted to reflect the new variable.
- If the original integral has limits \( a \) and \( b \), then after substituting \( u = g(x) \), the new limits become \( g(a) \) and \( g(b) \).

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#### Conclusion
Overall, Learning Objective FUN-6.D equips students with the skills needed to effectively apply substitution techniques in both indefinite and definite integrals. Mastery of these concepts allows for a deeper understanding of integration methods and enhances problem-solving abilities in calculus.

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