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Evaluate the given integral.$$\int \sqrt{4 x+1} 4 d x$$

$$\frac{1}{6}(4 x+14)^{3 / 2}+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

Campbell University

Harvey Mudd College

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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to integrate thes square root of four X plus 14. We can take advantage of the fact that we know how to integrate X to the one half. So let's rewrite this integral by introducing a use substitution and allowing you to be for X Plus 14. That way, do you? Over four is equal to DX, and we can rewrite this integral as 1/4 times the integral of you to the one half do you? And now we want to integrate you to the one half, which is the same as X to the one half because they're just dummy variables. And when we do that, we get that this is 1/4 times you to the one half plus one, which is three halfs, divided by one half plus one, which again is three halves plus C, which is equal to 1/4 times to over three times you to the three Halfs and U is equal to four X plus 14 so we can replace you by four X plus 14 to the three halves plus C. And we can simplify this a little bit further and write this as 1/6 times four X plus 14 to the three halfs. Oh, I see, and that completes the problem

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