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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.R.18
Capitolo 5, Problema 5.R.18

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.


―∫₄¹ 2ƒ(𝓍) d𝓍

Guida verificata passo dopo passo
1
Step 1: Recognize that the integral ∫₄¹ 2ƒ(𝓍) d𝓍 involves reversing the limits of integration. When the limits are reversed, the integral changes sign. Thus, ∫₄¹ 2ƒ(𝓍) d𝓍 = -∫₁⁴ 2ƒ(𝓍) d𝓍.
Step 2: Use the property of integrals that allows constants to be factored out. Specifically, ∫₁⁴ 2ƒ(𝓍) d𝓍 = 2∫₁⁴ ƒ(𝓍) d𝓍.
Step 3: Substitute the given value of ∫₁⁴ ƒ(𝓍) d𝓍 = 6 into the equation from Step 2. This gives ∫₁⁴ 2ƒ(𝓍) d𝓍 = 2 × 6.
Step 4: Combine the results from Step 1 and Step 3 to express the integral as -∫₁⁴ 2ƒ(𝓍) d𝓍 = -(2 × 6).
Step 5: Conclude that the integral ∫₄¹ 2ƒ(𝓍) d𝓍 can be evaluated using the steps above, but the final numerical result is not calculated here as per the instructions.

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Properties of Definite Integrals

Definite integrals have several key properties, including linearity and the ability to reverse limits. The linearity property states that ∫[a,b] (c * f(x)) dx = c * ∫[a,b] f(x) dx for any constant c. Additionally, reversing the limits of integration changes the sign: ∫[b,a] f(x) dx = -∫[a,b] f(x) dx. Understanding these properties is essential for evaluating integrals efficiently.
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Definition of the Definite Integral

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then ∫[a,b] f(x) dx = F(b) - F(a). This theorem allows us to evaluate definite integrals by finding the antiderivative, which is crucial for solving integral problems and understanding the relationship between the two operations.
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Fundamental Theorem of Calculus Part 1

Substitution in Integrals

Substitution is a technique used in integration to simplify the process of evaluating integrals. It involves changing the variable of integration to make the integral easier to solve. For example, if we let u = g(x), then the integral ∫ f(g(x)) g'(x) dx can be transformed into ∫ f(u) du, which may be simpler to evaluate. This concept is particularly useful when dealing with composite functions.
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