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Use the properties about odd and even functions to evaluate the given integral.$$\int_{-2}^{2}\left(3 x^{4}-2 x^{3}+4 x^{2}-2 x+1\right) d x$$

$$956 / 15$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 7

Substitution and Properties of Definite Integrals

Integrals

Missouri State University

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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Use the properties about o…

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Evaluate the integral usin…

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Write the integral as the …

01:12

All right. So the longest part of this problem be rewriting it. So you have some negative 2 to 2 of three x of the fourth minus two X cubed plus four X squared, minus two X plus one. I might just need to double check that. I copy that down correctly. Um, but what you can do because additions community, you can rewrite this as to separate functions. And then over here, I'm just gonna write the negative two x cubed minus two x. And the reason why you want to split it up like this is I split up the evens and the odds. Um, Well, what we learn is if we have the odd function, this integral will equal zero zero so we can disregard that. Whereas this integral because it's all evens. What we can actually do is just the integral from 0 to 2 and double it because even as symmetric with the y axis. So if we just find half that integral, we can double that answer. Whereas odd functions are symmetric with the origins of the area below will cancel out with the area above. So now all you have to do is find the integral equals in here, adding one to your exponent and divide by that new exponent. Add one to your exponent divided by your new exponent. Uh, not just X. We're going from 0 to 2. That the reason why you would do that is because taking two to the fifth power, uh, is 32 times three the 96 this to the third power eight tens, 4, 32 3rd plus two. But the reason why we do this plugging in zero here, here and here we get to I guess you can write out minus zero it's really easy math. Easier math. Um, so what you can then do I think I might even go to a calculator at this point because that's kind of complicated to add together. Um, and 56 would that be? I'm using a calculator. By the way, you should get 478 78 15th when you get the same denominator and add things together, Uh, and then when you multiply that, too in there, you'll get 956 15th. That should be your correct answer

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