If ∫²₋₂ 3ƒ(x) dx = 12, ∫⁵₋₂ ƒ(x) dx = 6, and ∫⁵₋₂ g(x) dx = 2, find the value of each of the following.

e. ∫⁵₋₂ ( ƒ(x) + g(x) ) dx
5

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Recall the property of definite integrals that states the integral of a sum is the sum of the integrals: \(\int_a^b (f(x) + g(x)) \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx\).

Identify the given integrals: \(\int_{-2}^5 f(x) \, dx = 6\) and \(\int_{-2}^5 g(x) \, dx = 2\).

Apply the property to the integral in question: \(\int_{-2}^5 (f(x) + g(x)) \, dx = \int_{-2}^5 f(x) \, dx + \int_{-2}^5 g(x) \, dx\).

Substitute the known values into the equation: \(\int_{-2}^5 (f(x) + g(x)) \, dx = 6 + 2\).

Combine the values to express the integral as a sum, which will give the final value once calculated.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Definite Integrals

Definite integrals have linearity properties, meaning the integral of a sum is the sum of the integrals, and constants can be factored out. For example, ∫[a to b] (f(x) + g(x)) dx = ∫[a to b] f(x) dx + ∫[a to b] g(x) dx. This allows breaking complex integrals into simpler parts.

Given Integral Values and Interval Consistency

When working with definite integrals, it is crucial to ensure the integration limits match when combining or comparing integrals. The problem provides integrals over specific intervals, so only integrals with the same limits can be directly added or manipulated.

Using Known Integral Values to Find Unknowns

Given certain integral values, you can use algebraic manipulation to find unknown integrals or expressions. For example, knowing ∫[a to b] f(x) dx and ∫[a to b] g(x) dx allows you to find ∫[a to b] (f(x) + g(x)) dx by summing the known values.

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