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Use the properties of integrals and the result of Example 3 to evaluate $ \displaystyle \int^3_1 (2e^x - 1) \, dx $.

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00:46

Frank Lin

00:42

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Missouri State University

Campbell University

Harvey Mudd College

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.

01:06

Use the properties of inte…

01:02

Use the result of Example …

01:59

Use properties of integral…

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Use a change of variables …

00:50

Yeah problem. 44. They want us to evaluate using properties of integral roles. The integral 1 to 3 to eat of the X men is one given that in a previous exercise. A few pages before this they work through an example and found this result. So properties of integral is here. This is going to be the integral from 1 to 3 of two. E to the X D x minus. The integral from 1 to 3 D X. That is to times the integral from 1 to 3. Either the X D x minus. The integral from 1 to 3 D X. Now the result that you see here, I already know that value so I can substitute that in. So this is going to be yeah, two. Mhm E cubed minus E minus. And this is just going to be three minus one. So when you do that integral is going to be So 3 -1. The reason you see this is, what does that function look like? This is like, well, one if y is equal to one from 1 to 3. Yeah, so here's one, here's one, here's three. What is the area of this rectangle? That is a one by three rectangle? Excuse me? one x 1 x two Rectangle. So it turns out this is just simply two E cubed minus E -2. Or if you factor to out of everything, EQ -E -1 is the final answer there.

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