The constant multiple and sum rules are your best friends when tackling indefinite integrals! The constant multiple rule states that the integral of a constant times a function is the constant times the integral of the function. Basically, you can pull constants out of integrals! The sum rule allows you to break down the integral of a sum of functions into the sum of individual integrals. These rules make integrating more complex expressions much easier.
Match the following terms with their definitions:
| Term | Definition |
|---|---|
| 1. Indefinite Integral | A. A function whose derivative is the original function |
| 2. Constant Multiple Rule | B. $\int kf(x) dx = k \int f(x) dx$, where k is a constant |
| 3. Sum Rule | C. The set of all antiderivatives of a function |
| 4. Antiderivative | D. An additive constant, C, added to an indefinite integral |
| 5. Constant of Integration | E. $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$ |
Complete the following sentences using the correct terms.
The integral of a constant times a function is the ______ times the integral of the function. This is known as the ______ rule. The integral of a sum of functions is the ______ of the integrals of each function, known as the ______ rule. Remember to add the ______ to your final answer when finding an indefinite integral.
Explain in your own words how the constant multiple and sum rules simplify the process of finding indefinite integrals. Give a specific example of an integral where both rules are applied.
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