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Use the properties about odd and even functions to evaluate the given integral.$$\int_{-1}^{1} \frac{x}{x^{2}+1} d x$$

$$0$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 7

Substitution and Properties of Definite Integrals

Integrals

University of Michigan - Ann Arbor

Idaho State University

Boston College

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 …

So when you leave this problem, we have the integral from negative one of one of X over X squared plus one. I know my students would struggle to understand that this is an odd function. Uh, but since the underlying theme in this is if this is an odd function and it is the the whole theme of it is that the area below the curve below the X axis will cancel with any area that's above the X axis. May I should even give you a visual. This function, you know, it's symmetric with the origin. Look, something like this. So the area below will cancel with the area above. So the answer is going to be exactly zero. Now for my students, I'd probably make them do the use substitution because I don't think they would be able to recognize that now, if you can. Then by all means, you know you. The answer is zero. We're already done just kind of showing you a more methodical approach to the problem. So you did that we would have the integral if I plug in negative. One squared is positive. One plus one is to I plug in the upper bound one squared is possible. One plus one is two and we have X over you times 1/2 X. Well, this is kind of nice, because if we start and stop at the same point, then this is confirming that we get an answer of zero. Otherwise, you can go through this whole process in your integral would be one half natural log of you from 222 So it'll be one half natural log of two minus natural log of two or one halftime zero, which is still zero. So no matter what you do, you're going to get an answer of zero zero zero. Mhm.

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