Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Evaluate the integrals.$$\int \frac{(x-2)^{2} \tan ^{-1}(2 x)-12 x^{3}-3 x}{\left(4 x^{2}+1\right)(x-2)^{2}} d x$$

$\frac{1}{4}\left(\tan ^{-1} 2 x\right)^{2}-3 \ln |x-2|+\frac{6}{x-2}+C$

Calculus 2 / BC

Chapter 8

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Integration Techniques

Campbell University

Oregon State University

Idaho State University

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

04:14

Evaluate the following int…

01:30

00:55

02:38

Evaluate the integrals.

11:02

Evaluate the integrals…

02:15

01:21

02:43

03:14

05:11

$$\text {Evaluate the foll…

and this problem. We're looking at the integral of X minus two squared Tanja Inverse of selects minus 12 X cubed minus three X All over four X squared, plus one times X minus two. Quantity squared, TX. Okay, so we have a real mess of a numerator and his army later, But we can kind of see a little bit of pattern going on here. So if we look at the first term, this X minus two quantities square and matches with this And if we look at the second term so this is the equivalent of negative three times four x cubed. This is the same as negative three x times four X squared, plus one. And we see that that matches with that. So if we break this fraction up into two pieces, we'll have a much easier problem to deal with. So kind of the idea with this problem issue? No. Here there's gonna be some kind of use substitution, which is something we learned in the past. And over here, we're gonna have some kind of partial fraction problem, which is something new. So we're just kind of taking old stuff we done and combining it with new things. So we're gonna have tangent inverse of she lacks over four x squared plus one, uh, minus three times X over. Um X menaced you. Quantity squared DX. Okay, so here we will need to do a u substitution where you is equal Teoh Tangent, inverse of two x. And the reason why we're going to do that is because when we differentiate to you, is can equal So by the changing role, we're gonna have shoe times shoot X squared plus one DX, which is to over four x squared plus one DX. And we have that for export post one D X right there. OK, but ask for this one that we're gonna have to use a partial fractions. Teoh, break this up a bit. So let's go ahead and we'll call this first integrate this first in July and the 2nd 1 I too. So when we solve her, I we're gonna have the enroll. Uh, 1/2 you. Do you that's gonna be equal, Teoh one over for you squared. Um, it's a constant. That's gonna be one force changes in verse of you quantity squared. No, no, no. Should be, uh ex change in degrees X quantity squared closer concept. Okay, so now we want to look at the 2nd 1 So we have the integral so negative three times integral of X over experience to quantity squared the X. So the top polynomial has a lower order than the bottom polynomial so we can eat Partial fractions Will have X over X minus two. What a square that's would be equal to some constant over X minus two, plus another constant over X minus two quantities where we can use the heavy side method cover up X menaced who and evaluate. So that's gonna give us B is equal. Teoh too, because we have X people. That, too, is the solution for X minus two you quantity squared equals zero. Okay, so then we multiply both sides by the denominator. To find a we're gonna have X is equal to 80 times X minus two, uh, plus two. So we have X minus two equals a times X minus two. That tells us that a is equal. The one so we have negative three. The integral of one over X minus to plus chew over wasn't too. Yeah, I was to two over X minus two Quantity squared dio numpty x. I want to integrate that. So what do we do? So this we're going to need to use of substitutions. Let's let the equal to expense to DV equals DX. So we'll have negative three Ln of absolute value of X minus two by that's three times the integral of shoe over the No. Two over V squared TV. So negative three. We need a constant because we integrated this already. So negative three Ellen absolute value of X minus two minus three. So we're gonna use the three times to we're going to use the power rule so we'll have a negative 11 over the wall, some constant. And what was the X minus two? So we have negative three l Emma X minus two minus plus six over We can write. That will better. So plus six over X minus to plus some constant Okay. And when we add those two roles together, what was the solution here it waas 1/4 tension in various quantity squared. OK, so then our total we're gonna have integral of X minus two square Tanja, members of two x minus 12 x cubed minus three x over four X squared plus one times X minus two quantity squared DX that's gonna be equal to 1/4 tangent inverse of shoe X quantity squared minus three l An absolute value of X minus chew uh, plus six over X minus two. Close. Some constant and that finishes their problems.

View More Answers From This Book

Find Another Textbook

In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

Evaluate the following integrals.$$\int \frac{2 x^{3}+x^{2}-2 x-4}{x^{2}…

Evaluate the following integrals.$$\int_{-1}^{1}\left(x^{3}-2 x^{2}+4 x\…

Evaluate the following integrals. $$\int_{-1}^{1}\left(x^{3}-2 x^{2}…

Evaluate the integrals.\begin{equation}\int \frac{x 2^{x^{2}}}{1+2^{x^{2…

Evaluate the integrals$$\int_{-1}^{1}\left(x^{2}-2 x+3\right) d x

Evaluate the integrals.$$\int_{-1}^{1}\left(x^{2}-2 x+3\right) d x$$

Evaluate the following integrals.$$\int \frac{d x}{x^{1 / 2}+x^{3 / 2}}$…

Evaluate the integrals.$$\int_{-2}^{2}(x+3)^{2} d x$$

Evaluate the following integrals.$$\int \frac{3 x^{2}+2 x+3}{x^{4}+2 x^{…

$$\text {Evaluate the following integrals.}$$$$\int \frac{x^{2}-4}{x^{3}…

01:45

In Exercises 35-64, use integration, the Direct Comparison Test, or the Limi…

00:39

Evaluate the integrals in Exercises $107-110$$\int_{1}^{e^{x}} \frac{1}{…

02:00

In Exercises $111-118,$ use logarithmic differentiation to find the derivati…

00:46

Evaluate the integrals in Exercises $93-106$

$\int 3 x^{\sqrt{3}} d …

01:16

08:20

In Exercises $11-16,$ use Euler's method to calculate the first three a…

00:28

Evaluate the integrals in Exercises $93-106$$\int x^{\sqrt{2}-1} d x$

06:00

Evaluate the integrals.$$\int \frac{e^{4 t}+2 e^{2 t}-e^{t}}{e^{2 t}+1} …

03:53

Theory and ApplicationsFind the absolute maximum and minimum values of $…

10:24

Social diffusion Sociologists sometimes use the phrase "social diffusio…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.