Explain why or why not Determine whether the following stateme...

Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(f) ∫ₐᵇ (2 ƒ(𝓍) ―3g (𝓍)) d𝓍 = 2 ∫ₐᵇ ƒ(𝓍) d𝓍 + 3 ∫₆ᵃ g(𝓍) d𝓍 .

Accepted Answer

Welcome back, everyone. Determine if the following statement is true. We're given two answer choices A as true and B's as false. Now, the statement that we want to analyze is the following. First of all, on the left hand side, it says that we have an integral from C to D of 4 H X plus 5K X all multiplied by DX. And we want to see if this is equal to 4 multiplied by the integral from C to D of H X X minus 5 multiplied by the integral from D to C of KXDX. So what we can do is simply apply properties of definite integrals and simplify. Let's split this integral into two parts. First, what we have integral from C to D of 4. H of XD X. Plus integral from C to D of 5KXDX. So we're basically applying the sum property, right? If we have a sum of two functions in our integrant, we can split our integral into two separate integrals. The second property that we're going to apply is the factorization of the constant so we can take out 4 and 5. So we get 4 integral from C to D of H of XDX plus 5 integral from C to D of K X D X. And finally, let's notice that on the right-hand side, we are swapping the limits of integration, right? We're looking at the integral from D to C of KFXDX. So we're going to leave our first integral as 4 integral from C to D. Of H of X X and for the second integral we're going to apply the Reverse interval property. So if we want to swap C and D, we get the integral from D to C of KFXDX and because we are swapping these limits, we have to add a negative sign in front. So we have -5 in front instead of positive 5 after swapping the limits. Now let's simplify this is equal to 4 integral from C to D of H of XDX plus minus 5, meaning we're subtracting 5 multiplied by the integral from D to C of K of XD X. And that we can see that this expression is equivalent to the expression on the right hand side, meaning the answer to this problem is a true. Thank you for watching.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.
(f) ∫ₐᵇ (2 ƒ(𝓍) ―3g (𝓍)) d𝓍 = 2 ∫ₐᵇ ƒ(𝓍) d𝓍 + 3 ∫₆ᵃ g(𝓍) d𝓍 .

Key Concepts

Properties of Definite Integrals

Linearity of Integration

Continuity of Functions

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