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Evaluate the given integral and check your answer.$$\int \sqrt[3]{x} d x$$

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

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 1

Anti differentiation - Integration

Integrals

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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.

01:58

Evaluate the given integra…

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Evaluate the integrals.

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Evaluate the integral.…

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all right, This is just algebra manipulation as well as learning a new trick. Um, that when you're doing the integral, the cube root of X dx, what I would do is rewrite this as the integral of X to the one third power DX. And this is a trick. You know, this is a true statement. We learn that in algebra eso the reason why I bring that to your attention is now having a rational exponents makes it easier to add one to your exponents. And then this is the process for the and hydro driven, by the way, and then multiply by the reciprocal have you new export. So we're ready to find this derivative anti derivative. Um, if you need help adding one toe one third, I would suggest rewriting it with same denominator is three thirds is equal to one. So one third plus three thirds is four thirds and then multiply by the reciprocal four thirds, which is three force. The only thing I didn't write down is that you also have to write plus C in this and this is your final answer, and you can check your work as they ask you to do by taking the derivative of that. Um so we've already found the answer and just talking about how to check your work. If you take the derivative and you bring that exponents front well, the force will cancel on the threes of cancel, and you multiply. When you subtract one from four thirds, you get one third, and the directive of a constant is zero. That's why we need to have a constant here eyes. There's an infinite number of equations whose derivative is this, which is just what we checked. So, like I said, circled in green is your final answer on blue work is just checking work.

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