Use symmetry to explain why.
∫⁴₋₄ (5𝓍⁴ + 3𝓍³ + 2𝓍² + 𝓍 + 1) d𝓍 = 2 ∫₀⁴ (5𝓍⁴ + 2𝓍² + 𝓍 + 1) d𝓍 .

검증된 단계별 안내

Step 1: Recognize that the integral is over a symmetric interval [-4, 4]. Symmetry in integrals often simplifies calculations, especially when the integrand has specific properties like even or odd functions.

Step 2: Break down the integrand into individual terms: 5𝓍⁴, 3𝓍³, 2𝓍², 𝓍, and 1. Analyze each term to determine whether it is an even function or an odd function. Recall that even functions satisfy f(-𝓍) = f(𝓍), while odd functions satisfy f(-𝓍) = -f(𝓍).

Step 3: Identify the symmetry of each term: 5𝓍⁴ and 2𝓍² are even functions, while 3𝓍³ and 𝓍 are odd functions. The constant term 1 is also even because it does not depend on 𝓍.

Step 4: Use the property of integrals over symmetric intervals: The integral of an odd function over [-a, a] is zero because the positive and negative contributions cancel out. Therefore, the terms 3𝓍³ and 𝓍 do not contribute to the integral over [-4, 4].

Step 5: Rewrite the original integral by excluding the odd terms and focusing only on the even terms. This simplifies the integral to ∫⁴₋₄ (5𝓍⁴ + 2𝓍² + 𝓍 + 1) d𝓍 = 2 ∫₀⁴ (5𝓍⁴ + 2𝓍² + 𝓍 + 1) d𝓍, leveraging the symmetry of the even functions over the interval.

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주요 개념

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Symmetry in Functions

Symmetry in functions refers to the property where a function exhibits identical behavior on either side of a central point, typically the y-axis for even functions or the origin for odd functions. For example, a function f(x) is even if f(-x) = f(x), and odd if f(-x) = -f(x). This property can simplify the evaluation of integrals, particularly over symmetric intervals.

Definite Integrals

A definite integral calculates the net area under a curve defined by a function over a specific interval [a, b]. It is represented as ∫_a^b f(x) dx and provides a numerical value that represents this area. Understanding how to manipulate definite integrals, especially with respect to symmetry, is crucial for simplifying calculations.

Properties of Integrals

Properties of integrals include various rules that allow for the manipulation and evaluation of integrals. One important property is that the integral of an even function over a symmetric interval [-a, a] can be expressed as twice the integral from 0 to a. This property is essential for simplifying integrals involving symmetric functions, as seen in the given equation.

Use symmetry to explain why.∫⁴₋₄ (5𝓍⁴ + 3𝓍³ + 2𝓍² + 𝓍 + 1) d𝓍 = 2 ∫₀⁴ (5𝓍⁴ + 2𝓍² + 𝓍 + 1) d𝓍 .