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.

a. ∫²₋₂ ƒ(x) dx

Verified step by step guidance

Recall the linearity property of integrals: for any constant \( c \), \( \int_a^b c f(x) \, dx = c \int_a^b f(x) \, dx \).

Given \( \int_{-2}^2 3f(x) \, dx = 12 \), use the linearity property to express this as \( 3 \int_{-2}^2 f(x) \, dx = 12 \).

To find \( \int_{-2}^2 f(x) \, dx \), divide both sides of the equation by 3, resulting in \( \int_{-2}^2 f(x) \, dx = \frac{12}{3} \).

Simplify the right-hand side to express the integral \( \int_{-2}^2 f(x) \, dx \) in terms of a numerical value (do not calculate the final number here).

This gives you the value of \( \int_{-2}^2 f(x) \, dx \) based on the information provided.

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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 properties such as linearity, which allows constants to be factored out and integrals to be split or combined over intervals. For example, ∫_a^b c·f(x) dx = c·∫_a^b f(x) dx, and ∫_a^c f(x) dx + ∫_c^b f(x) dx = ∫_a^b f(x) dx.

Linearity of Integration

Integration is a linear operation, meaning the integral of a sum is the sum of the integrals, and constants can be factored out. This helps in manipulating given integrals to find unknown values by breaking down or combining integrals.

Using Given Integral Values to Find Unknowns

Given specific integral values over certain intervals, you can use properties of integrals to solve for unknown integrals over subintervals or related functions by setting up equations and applying algebraic manipulation.

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