Use symmetry to explain why.
∫⁴₋₄ (5𝓍⁴ + 3𝓍³ + 2𝓍² + ...

Question

Use symmetry to explain why.

∫⁴₋₄ (5𝓍⁴ + 3𝓍³ + 2𝓍² + 𝓍 + 1) d𝓍 = 2 ∫₀⁴ (5𝓍⁴ + 2𝓍² + 𝓍 + 1) d𝓍 .

Accepted Answer

Welcome back everyone. Using symmetry, determine whether the following equation is true or false that the integral between -5 to 5 or 4 X 6 plus 6 X 5 plus 3X2 + 2X + 7 or with respect to X equals 2 multiplied by the integral between 0 and 5 of 4 X 6 plus 3 X 2 + 7, all with respect to X. A says it's true. B says it's false. Now what do we know that can help us to determine the validity of our equation, how true our equation is? Well, let's try to start thinking about each of our terms. What do we know? Well, notice that in our problem here we, it's made up of even terms, OK. And it's made up of odd terms. What do we know about both? Well, recall, OK. That if a function is even. OK, then that means F of negative X is equal to F of X, OK? And evaluating the definite integral for that, then. For an odd function over the integral from A to negative A, OK, it's going to be equal to 2 multiplied by the integral of the same function from 0 to A, OK? Next, we know that if a function is odd. OK, then F of negative X is going to be equal to negative F of X, which means that this definite integral from negative A to A is going to be equal to 0, OK? So how can that help us to determine if the statement is true or false? Well, if we were to apply it, OK, applying this to our integral, OK. Then This tells us that, well, let me rewrite our integral here. For our terms, 4 X 6 + 6 X 5 + 3 X squad + 2 X + 7. Then we know because it's a sum we'll be applying this in this definite integral to each of our terms and once we do that based on what we just figured out, then 6 X to the 5th and 2 X, both odd terms will integrate to zero over this inteval. On the other hand, 4 X to the 63 X squad, and 7 will all combine, OK, and their integrals will double based on our first statement. It will double for the same a similar interval from 0 to A. in this case from 0 to 5. So now if we combine those terms, combine all of those even terms, then we're going to end up. With 2 multiplied by the integral from 0 to 5. Of 4 X to the 6 plus 3 X squared. Plus 7. All with respect to X. This matches the right side of the original equation. Thus, the equation is true. Symmetry has proven it. Therefore, A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Use symmetry to explain why.
∫⁴₋₄ (5𝓍⁴ + 3𝓍³ + 2𝓍² + 𝓍 + 1) d𝓍 = 2 ∫₀⁴ (5𝓍⁴ + 2𝓍² + 𝓍 + 1) d𝓍 .

Key Concepts

Symmetry in Functions

Definite Integrals

Properties of Integrals

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