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Evaluate the integral.
$ \displaystyle \int^3_{-2} (x^2 - 3) \,dx $
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00:42
Frank Lin
00:46
Amrita Bhasin
Calculus 1 / AB
Chapter 5
Integrals
Section 4
Indefinite Integrals and the Net Change Theorem
Integration
Oregon State University
University of Nottingham
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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Okay, so this is definitely a girl. We're going to take the anti derivative of this term inside here X squared minus three. And then apply the net change there. So we can start with the anti derivative of X squared X cubed over three. and then for three just That would be three x. Right upper bound of three. Lower negative too. So now we can write this as. So your plan that changed theorem plugging three for X. We get three cubes over three -3 times three. It's the first term. And then subtract negative two cubed over three. Why is three times negative two sort of valuing this? We can get three cubed just 27 over three Most three times 3 is nine from this. We subtract negative two cubed. Um She'll be negative 8/3 -3 times. To give to some minus negative six which would just be plus six. But now 27/3 is nine so 9 -9. This first term will cancel out. So we get negative negative eight over three, then Get a common denominator. We can change this six into 18 over three, so 18/3 minus 8/3 will be 10/3 to add that negative side in front. Get negative 10/3 as our answer.
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