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
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Guida verificata passo dopo passo

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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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.

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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.

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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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