Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.
(a) ∫₀³ 5ƒ(𝓍) d𝓍

검증된 단계별 안내

Step 1: Recognize the property of integrals that allows constants to be factored out. Specifically, for any constant c and function f(x), ∫ₐᵇ cƒ(𝓍) d𝓍 = c ∫ₐᵇ ƒ(𝓍) d𝓍.

Step 2: Apply this property to the given integral ∫₀³ 5ƒ(𝓍) d𝓍. Here, the constant 5 can be factored out, resulting in 5 ∫₀³ ƒ(𝓍) d𝓍.

Step 3: Substitute the value of ∫₀³ ƒ(𝓍) d𝓍 provided in the problem, which is 2.

Step 4: Multiply the constant 5 by the value of the integral ∫₀³ ƒ(𝓍) d𝓍 (which is 2) to complete the evaluation.

Step 5: The result of the integral ∫₀³ 5ƒ(𝓍) d𝓍 is obtained by performing the multiplication in Step 4.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Properties of Integrals

The properties of integrals, particularly the linearity property, state that the integral of a constant multiplied by a function can be factored out. This means that ∫a^b kƒ(𝓍) d𝓍 = k∫a^b ƒ(𝓍) d𝓍, where k is a constant. This property simplifies the evaluation of integrals by allowing constants to be taken outside the integral.

Definite Integrals

Definite integrals represent the signed area under a curve between two limits. The notation ∫a^b ƒ(𝓍) d𝓍 indicates the integral of the function ƒ(𝓍) from the lower limit a to the upper limit b. The result of a definite integral is a number that quantifies this area, which can be positive, negative, or zero depending on the function's behavior over the interval.

Additivity of Integrals

The additivity property of integrals states that the integral over an interval can be split into the sum of integrals over subintervals. Specifically, ∫a^c ƒ(𝓍) d𝓍 = ∫a^b ƒ(𝓍) d𝓍 + ∫b^c ƒ(𝓍) d𝓍 for any point b between a and c. This property is useful for evaluating integrals over larger intervals by breaking them down into smaller, manageable parts.

Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.(a) ∫₀³ 5ƒ(𝓍) d𝓍