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Evaluate the integral.

$ \displaystyle \int^{4}_{1} \biggl( \frac{4 + 6u}{\sqrt{u}} \biggr) \,du $

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01:10

Frank Lin

00:37

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 4

Indefinite Integrals and the Net Change Theorem

Integration

Missouri State University

Campbell University

University of Nottingham

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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We're doing the inner roll from 1-4 of this function. Uh Four plus six. You all over the square root of you? Um I'll talk about what I'm gonna do next. So it be in our best interest if we rewrote this because we're dividing by a mono mule if we divide each piece. Uh And it makes sense to write it as to the negative one half power uh negative because it's in the denominator and then the one half for the square root. Uh And then plus then divide this piece. Um And then here if you think of this as you do the first over you to the one half, you subtract the exponents. It's to the positive one half power, do you? And now we can follow our rules that we have learned where we add one to the exponent for the anti derivative. And you multiply by the reciprocal. Um So there is typical for one half is two and four times two gives me eight. Uh The same thing with the one half plus one is three. Has multiply by the reciprocal. Uh Maybe six times two it was just 12 to 5 by three is four. And you can double check this is right by taking the anti derivative of that. So now what we have to do is plug in our upper bound, which what this means is the square root of four because we're plugging in for. So it's two and then this is the square root of four which is to cuba, which gives me eight. And then we have to subtract off. What's nice about plugging in? One is one to any powers just one. So we're looking at a plus for their. So as I'm doing this I'm seeing six Sorry, attempts to 16 Plus 32 which would give me 48 uh minus 12 Or a final answer of 36. And you might need a calculator to verify that. But this is the correct answer. Mhm. Yeah.

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