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Use the result of Example 3 to evaluate $ \displaystyle \int^3_1 e^{x + 2} \, dx $.

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00:52

Frank Lin

00:26

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Campbell University

Oregon State University

Baylor University

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.

00:50

Use the result of Example …

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$$\text { Use the resu…

01:06

Use the properties of inte…

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02:31

Evaluate the integral.

03:13

Use a substitution to eval…

02:10

Evaluate the integral.…

Yeah, example here integrate the definite integral 1 to 3 of E to the X plus two. Given the fact that we've already worked the integral from 1 to 3 of E to the X is E cubed minus E. So we can rewrite this integral using properties of exponents. So this is the integral from 1 to 3 of E to the X E squared dx E squared is simply a constant. So this is E squared. The integral from 1 to 3 E to the X dx. That is precisely the result that you see right here. So this answer just turns into e squared E cubed minus E. Um you could factor one mori out of this and this is going to be what E cubed um E squared minus one or simply Each of the 5th minus E cubed. Mhm.

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