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

$$\frac{4 x^{3}}{3}+2 x^{2}+x+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 1

Anti differentiation - Integration

Integrals

Campbell University

Harvey Mudd College

Idaho State University

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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Evaluate the given integra…

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Evaluate the integral and …

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Evaluate the given definit…

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Evaluate the integral.…

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okay. It's very important that you understand the simplification in this problem. Um, because that will impact your answer. Eso What I would make my students do is rewrite. This problem is understanding that each piece needs to be cute, so that means Three Cube that's three times three times three gives me 27 x cubed DX. So I would make my students do that before they do the process of the anti directive, which just add one to the exponents and then divide by your new exponents. And that's how we do the anti derivative. So all you're going to see me do is add 123 so it becomes four and then divide by four. That 27 is still there now. Sometimes this reduces, but that's it. Um, don't forget about a Plus C, though, and why do we need a plus? C is there's an infinite number of correct answers because there's an infinite number of equations, depending on what things constant is because it could be anything where the derivative of Constant is zero. But we don't write down plus zero up here. Eso That's why we need that plus c so to check your work. It's asking you to find the derivative of our answer. So is it true that 27 Force X to the force derivative is 27 Execute? Yes, it ISS so derivative of a constant is zero. So we're good.

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