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

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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