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Evaluate the given integral and check your answer.$$\int \frac{1}{x^{3}} d x$$

$$\frac{-1}{2 x^{2}}+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 1

Anti differentiation - Integration

Integrals

Missouri State University

Campbell University

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.

01:32

Evaluate the given integra…

01:28

01:04

Evaluate the integral and …

01:33

02:34

All right, So this problem just involves a little bit of algebra manipulation. One over. Execute ds eso don't do the anti director of yet. Rewrite this with a negative exponents. We'll just reminder that if you're in the denominator, that that's equal to having a negative exponents in the new Marie eso. The reason why I have students do this is your two step process to doing the anti derivative is add one to the exponents on divide by your new excellent or multiply by the reciprocal. I've heard teachers say that, which is also correct. Um, so what am I looking at? Well, if I add one to negative three, just scratch work. Native three plus one is equal to negative two. And then I need to divide by that new X one, which is the same as one divided by negative, too. I guess I'm actually multiply mother cyclical. But then you also need this plus C into the problem. Um, now, since we started with positive exponents, I would anticipate my teacher saying, Hey, you have to rewrite this with positive exponents. So that's sticking the X to the native second power back into the denominator So this is how I would write my answer. Now the directions also ask you to check your work. Now, if I'm checking my work, I would have actually gone back to this stuff and then taken a derivative of that. So and blue is me checking my work of negative one half next to the next second power plus c. Well, if I did that derivative, I would bring that in front. Will. Negative One half of negative two is positive. One. Subtract one from that exponents. Negative three. The derivative of a constant zero. That's why we need a constant in there because we don't know what that constant could be. And excellent of third powers. The same thing is up here. So my answer and green is correct. Circling green acquisitions.

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