Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

Make a substitution to express the integrand as a…

05:59

Question

Answered step-by-step

Problem 51 Medium Difficulty

Make a substitution to express the integrand as a rational function and then evaluate the integral.

$ \displaystyle \int \frac{dx}{1 + e^x} $


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

JH
J Hardin
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by J Hardin

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Related Topics

Integration Techniques

Discussion

You must be signed in to discuss.
Top Calculus 2 / BC Educators
Grace He
Catherine Ross

Missouri State University

Heather Zimmers

Oregon State University

Samuel Hannah

University of Nottingham

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

Join Course
Recommended Videos

08:52

Make a substitution to exp…

08:23

Make a substitution to exp…

05:06

Make a substitution to exp…

03:49

Use substitution to conver…

02:22

Use substitution to conver…

05:17

Make a substitution to exp…

01:04

Use substitution to conver…

Watch More Solved Questions in Chapter 7

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75

Video Transcript

Let's try the substitution you equals Eat of X. The goal here is to use a substitution to rewrite the inner grand one over one plus needed the X as a rational function. So taking this to the U Do You is here The X T X, which I could write, is you DX and then divide by you. Take it, do you over you equals DX. So in the original integral, I can replace DX with do you over you and then in the denominator, we just have one plus you. So let's clean this up a little bit. This is just one over and then we have you one plus you to you. And that's a rational function that they asked for. So taking this rational function, let's go ahead and do partial fraction to composition, using what the author calls case one you distinctly of actors. Let's go and multiply both sides by that denominator on the left, and then let's go ahead and simplify by pulling out of you from the left hand side. The constant term is one on the right hand side. It's a so those are equal and because on the right we have a plus B in front of the U next to the U. But on the left, there is no you. So the coefficient in front of you, you must be zero. So solving this for me, we get B equals negative one and then we could plug These values in for Ambien are partial fraction and then we take this and this is the term that we're replacing the fraction with. And then we integrate. Let's go to the next place to do that. They was one so integral over you, minus one for B. So we pull out the minus one plus you, do you the first general natural log? Absolute value. Second rule, absolute value. One plus you. Now that one Plus he was bothering. You feel free here to do a use up. Let's say w equals you plus one, and that should help you evaluate the integral with that said the last step here. Replace you with how it's to find needed X. You could drop the absolute value here since you two, the exes always positive. And you could also drop it here because either the X plus one is positive, the last possible thing we can do here is to use the fact that Ellen X in need of X r inverse functions. So remember, if you have in Versace and you composed them together, you always get X. And that's exactly what we have here. Natural Log is being composed with either the ex, so that is just X. And then everything else is the same, and that's your final answer.

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
151
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
68
Hosted by: Alonso M
See More

Related Topics

Integration Techniques

Top Calculus 2 / BC Educators
Grace He

Numerade Educator

Catherine Ross

Missouri State University

Heather Zimmers

Oregon State University

Samuel Hannah

University of Nottingham

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

Join Course
Recommended Videos

08:52

Make a substitution to express the integrand as a rational function and then ev…

08:23

Make a substitution to express the integrand as a rational function and then ev…

05:06

Make a substitution to express the integrand as a rational function and then ev…

03:49

Use substitution to convert the integrals to integrals of rational functions. T…

02:22

Use substitution to convert the integrals to integrals of rational functions. T…

05:17

Make a substitution to express the integrand as a rational function and then ev…

01:04

Use substitution to convert the integrals to integrals of rational functions. T…
Additional Mathematics Questions

00:54

A data set of 30 values has a mean of 6.2. What is the sum of the values?

01:48

6 typist working 5 hours a day can type the manuscript of a book in 16 days.…

01:22

Learning Task 3: Solve the given problems using 4-steps in solving word prob…

02:47

Michelle has an average score of 70 for three tests. What must she score on …

01:11

Which of the following is a well-defined set?
A. {favorite colors}
B. …

00:48

Which of the processes below will complete the given statement?“ To add two …

02:02

if a set of data has a mode, then must the mode be one of the numbers in the…

03:26

Which of the following is the mean absolute deviation for the set of data? 2…

01:48

Janetta brought to school 4/5 of a cake which she and her 3 friends shared e…

00:22

if you earn P40,000 Per month and you spend P19,000 per month write an equat…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started