Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Definite Integrals
Definite integrals have several key properties, including linearity, which states that the integral of a sum is the sum of the integrals. This means that β«(f(x) + g(x)) dx = β«f(x) dx + β«g(x) dx. Additionally, the integral from a to b can be expressed as the negative of the integral from b to a, i.e., β«_a^b f(x) dx = -β«_b^a f(x) dx.
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Definition of the Definite Integral
Change of Limits in Integrals
When evaluating integrals, changing the limits of integration affects the sign of the result. Specifically, if you reverse the limits of integration, the value of the integral becomes negative. For example, β«_a^b f(x) dx = -β«_b^a f(x) dx, which is crucial when evaluating integrals with limits that are not in increasing order.
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Linear Combination of Functions
A linear combination of functions involves adding or subtracting functions multiplied by constants. In the context of integrals, if you have a function like (f(x) + 2g(x)), you can evaluate the integral of this combination by integrating each function separately and applying the constants accordingly. This property simplifies the evaluation of integrals involving multiple functions.
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