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Evaluate the given integral and check your answer.$$\int \sqrt[4]{x^{5}} d x$$

$$\frac{4}{9} x^{9 / 4}+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 1

Anti differentiation - Integration

Integrals

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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.

01:56

Evaluate the given integra…

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Evaluate the integral.…

02:20

Evaluate the integrals.

06:32

02:32

Find the indefinite integr…

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Evaluate the definite inte…

00:43

Yeah, all right. Our task is to find the integral of the fourth root of X to the fifth DX. And what I always have my students do is just rewrite this with rational exponents. The reason why that's hopeful is then adding one to that exponents is easier. And then what you also need to do is multiply by the reciprocal of that new experiments. Eso if you were ready to find the anti derivative already, uh, like if your struggle with adding 1 to 5 force just get the same denominator denominator being four. So four force is equal to one s 05 plus four is nine. The denominator stays the same force, multiply by the reciprocal of 94 so four nights and don't forget about plus seat. And this is your correct answer. And when they ask you to check your work, they're just saying, Take the derivative of that answer. And when you do that, you'll see why we undo by adding and multiplying. Um, because when you bring that nine force in front, it cancels out with the nine and the four. When you subtract one from your exponents, you get back to that five force and the derivative of a constant zero. And you see how we get back up there? Which is why we have a constant because we don't know what this concept is in the derivative of any concept is going to be zero. So that's why we need this is part of our answer circled in green.

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