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Evaluate the integral.

$ \displaystyle \int^2_{1} (4x^3 - 3x^2 + 2x) \,dx $

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00:46

Frank Lin

00:36

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 4

Indefinite Integrals and the Net Change Theorem

Integration

Oregon State University

Harvey Mudd College

University of Nottingham

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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Evaluate the indefinite in…

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Evaluate the integral.…

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01:24

we have to do the integral from 1 to 2 of this function for X cubed minus three X squared. Uh plus two x. And uh I think they did this on purpose or the anti derivative, you add one to your exponents and then you divide by your new expo before divided by four will cancel each other out. And you can double check this is correct because the derivative of X to the fourth is four X cubed. And the same thing happens for the next two as well. Uh When you add one to the extra notes, cancel with that 33 divided by three is one and something with this Uh had one to the expo and to divide by two cancelled out and then that's going from 1-2. So now what you need to do is plug in your upper bound so that's two to the 4th -2. Cute plus two squared. And then plug in your lower bound one to the fourth minus one cube plus one squared. And what's nice about all of this at least? I think it's nice. Um 24816 as these are very easy numbers to work with to Cuba's eight Plus four and then one day any power is just one. Um So we're looking at one minus one is zero plus one is one. Uh So 16 minus eight is eight plus four is 12 minus one is 11. Your final answer? Mm

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