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

(a) Find the partial fraction decomposition of th…

05:40

Question

Answered step-by-step

Problem 69 Hard Difficulty

(a) Use a computer algebra system to find the partial fraction decomposition of the function
$$ f(x) = \frac{4x^3 - 27x^2 + 5x - 32}{30x^5 - 13x^4 + 50x^3 - 286x^2 - 299x - 70} $$
(b) Use part (a) to find $ \int f(x) dx $ (by hand) and compare with the result of using the $ CAS $ to integrate $ f $ directly. Comment on any discrepancy.


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

Anna Marie Vagnozzi

Campbell University

Caleb Elmore

Baylor University

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

05:40

(a) Find the partial fract…

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

suffer party. Let's use a computer algebra system to find the partial fraction to composition. So answer this fraction into a computer algebra system. So here's Wolfram Alpha with input, the freshen. And then if we scroll down, we see a partial fraction to composition that's highlighted here. So let's go ahead and write all this on the white board. So for her, eh, we can rewrite f is the following. So we do have a lot of writing here with the large numbers. So this is using the computer algebra system. Tau. Give us the partial fraction to composition. So there's one term so that we have six, six a three, two, three, two worthless one also minus nine, four, three, eight, eight zero one, five, five And then we have one more term after this. So this is just for party here in one more term, twenty four one zero one one zero for eight, seven nine. Finance clothes, too. Now, the next part will be to actually God and integrate this. So the first integral is going to be the most difficult because of the quadratic. So what? To go ahead and complete this wear here you can write this as experts a have square, the whole thing and then radical nineteen Over too square. So we just completed the square there. Let's spend a few moments just on this first Integral. The other three hundred girls will just involve natural log And you could do them all by U substitution. So here let me take this first freshen and then just go ahead and write this. So what I'LL do here is I see that I haven't x plus one half on the bottom. So I'll rewrite this x It's X plus one half But by doing so, I have to subtract. So we have to to let me let me take a step back here That's two X plus one half and then we still have the plus forty eight thousand. But since I just added in this extra half, I have to make up for by subtracting one half of this number and then now I'm you just split this into two fractions. I still have that two hundred sixty thousand out there And now, after I completed the square, I have this and then route one nine over two square that as well So that's the first in immoral and the same denominator will show up in the infraction. Little sloppy here. And then we could even put a d s up here because we're integrating now was break this into two parts. So here we should take you to just be explosive. Have take it to be the denominator there without this number in the front. And then you over too. Expose the have d x so that in a rule, becomes it's then from our use of we have one half one over you We'LL see you there so this becomes it's a four up there in the front for nine Then we're natural log of you and then we could replace you with and swear was X plus five was put that in absolute value. So this right here is just Ellen ofyou and I just replace you with definition here on the simple side. So that's the first in a brawl. So remember, on the previous page we have foreign rule severe with we're just dealing with the most difficult undergo here that we circled in red. We split in the institute hearts. We just found one of the parts. We're one more parts of fine. And then we'LL simplify the other three simple intervals. Position of rule. Let's do trace. Um so go ahead and evaluate this. Using this truths up in the same manner that I did over here using the use home in this in a roll he comes seven five, seven seven two, two hundred sixty thousand We have that nineteen on the bottom and then our tan for this is from the tricks up two X plus one and then on the bottom. Radical nineteen. So the first most complicated in a girl that we had we just broke that into two parts. Now we also so we'll just add those together at the very end And then we had the second and rule six, six eight over three to three two x plus one. You're free to do a use up here. Then the next interval. The third one, who's a eighteen to be X minus seven gs. So go ahead and use up here as well. You forced three x minus seven You get three one four six He's zero one one one five five. Natural log three x minus seven and then for the final in a rule. So here's our last Enbrel. Then we find we don't get him at everything all of the end. So go ahead and use of here equals five plus two. Simplify this integral to the natural world. By that plus two, the absolute value and other final answer was It's just the interval, the sum of the two hundred girls that we had in the previous age, then minuses in general, minus the green in a brawl. And then plus this last read and mumble. So let me go to the next page to write that. So our final answer it's And then after we write this down will go head backto Wolfram to the computer algebra system, and we'LL see if they're any differences in how our final answers there. My guess is that they just won't have absolute value inside the natural, and we'LL check it momentarily. Let's keep writing this answer down, adding, It's attracting all these in a girls together, so that's the sum that's the most complicated. And a girl. That's what we had used to usurp amended tricks up. Then the next, in a brawl them we subtracted there following in a role as well. So it's a one five five ln three x minus seven and then the final interval, we added that was for a two two. So this is our last expression here. Natural log five x plus two. And finally we could go ahead and add in that constant of integration. See? So this whole page, this whole y borders are final answer and let's go ahead and compared to Wolfram. So no, the constants not know. Also, we have absolute value here for all of our loves. If you go back to Wolfram, our computer algebra system, let's scroll down and it gives us the indefinite in the rule of our friendship. It's just write it in a different order, but it is the same fraction. And if you look inside, the longs hereby log, They mean natural other of them, as they mentioned on the side. But they do not have absolute values, but otherwise our answers are the same. So 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
142
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

Anna Marie Vagnozzi

Campbell University

Caleb Elmore

Baylor University

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

05:40

(a) Find the partial fraction decomposition of the function $$ f(x) = \frac{12…
Additional Mathematics Questions

06:08

'Which number line shows the solution of -3x + 6 > 12?
Which numb…

06:08

'Which number line shows the solution of -3x + 6 > 12?
Which numb…

06:08

'Which number line shows the solution of -3x + 6 > 12?
Which numb…

02:24

'For the parallelogram, m angle QRP = 32 and m angle PRS = 84. Find m a…

02:18

'CAN SOMEONE HELP WITH NUMBER 9 PLZZ
5. AJKL and APQR
Determine w…

01:33

'Which graph shows a negative acceleration?
Position (m)
Time (s)…

00:52

'Help me Plz if you answer correctly I will choose u as brainliest
F…

02:25

'What fraction of an hour is 18 minutes.
f 10
What fraction of an…

01:06

'Which choice is equivalent to the product below?
Question 4 of 10<…

05:18

'what’s the answer to the riddle?? please help :(
RIDDLE: LINEAR-QU…

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