Problem

Integrate each of the given functions. $$\int e^{…

Problem 2 Easy Difficulty

Make the given changes in the indicated examples of this section and then solve the given problems.
In Example 4, change $e^{x+1}$ to $e^{x-1}$ and then integrate.

Answer

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Basic Technical Mathematics with Calculus

Chapter 28

Methods of Integration

Section 3

The Exponential Form

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Video Transcript

Okay so we're gonna integrate this function which is equal to four times E. To the three X power minus three times E. To the X power divided by E. Raise the x minus one power. And so the way that we can do this is we're going to break this um fraction up into two fractions and then we're going to integrate both of those individually. So we have the integral four E 23 X Divided by E. to the X -1 power. And this is minus the integral of three E. To the X power divided by E. To the X -1 power. And so what we can do now is since we have um the same base here, E. T. In the numerator as well as the denominator we can minus the exponent in the numerator. Or sorry the exponent in the denominator from the exponent in the numerator. So this would be equal to the integral four times e three x minus x minus one. Mhm. And then we have D X minus the integral of three E. To the X power minus x minus one. And we also have a dx here. So three X- X -1. This is going to be equal to the integral of for times E. To the two X plus one. And this is D. X. And then we have minus the integral three times E. To the first power, D. X. So E to the two X plus one power. That's equal to The integral of four types, E times E To the two x power. And so what we can do now is we can take out this four. And that's what I'm going to do over here. So this is gonna be minus the -3 times e. multiplied by the integral of just dx. So we're going to take out this four E on this integral. So this is going to be equal to four. He times the integral E. to the two x -3. E times the integral D. X. And I forgot to dx here forgot to dx here as well. Um And so now we can find these integral. So the integral of each of the two X. Um if we let U equal to X Then the U. is equal to two. So this would be equal to for e times the integral of one half. Since to use equal to two we have to divide by two to make um are integral equal to each other, multiplied by E. Race the power of you multiplied by D. You let me have minus three E. multiplied by integral of DX. And so now the integral of E. To the U. Is just E. To the U. So this would be equal to one half times four is too, so would be equal to two E. Multiplied by E. To the U. Plus C. And then we have minus um the integral of dx is just X. So this is gonna be three E times X. We also have plus C. Here. Um And so what we can do now is distribute this two Ee after we plug in back what U. Is equal to, so U. Is equal to two X. So this is equal to two e. Multiplied by E. 2 2 x. And then we have -3 e. Times X. And then we have plus C. Since our um constant here is just gonna be one constant since what we started with was one integral of this function. So we're only gonna have one constant of integration. And so what we can do now is we can um multiply the E by the each of the two X. So this would be to E. To the two X plus one minus three, E. X. Plus C.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Problem

Integrate each of the given functions. $$\int e^{…

Problem 2 Easy Difficulty

Make the given changes in the indicated examples of this section and then solve the given problems.
In Example 4, change $e^{x+1}$ to $e^{x-1}$ and then integrate.

Answer

Related Courses

Basic Technical Mathematics with Calculus

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Watch More Solved Questions in Chapter 28

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Kayleah T.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

Basic Technical Mathematics with Calculus

Grace H.

Kayleah T.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Grace H.

Kayleah T.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus