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

Verified step by step guidance

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.
2. Step 2: Use the property of integrals that allows constants to be factored out. Specifically, ∫₁⁴ 2ƒ(𝓍) d𝓍 = 2∫₁⁴ ƒ(𝓍) d𝓍.
3. 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.
4. Step 4: Combine the results from Step 1 and Step 3 to express the integral as -∫₁⁴ 2ƒ(𝓍) d𝓍 = -(2 × 6).
5. 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.

Key Concepts

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.

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.

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.