Evaluate Integral 4X/(X^3+X^2+X+1), Evaluate The Integral

Note that $x^3+x^2+x+1=(x+1)(x^2+1)$. We use partial fracions. So we try to find constants $A,B,C$ such that$$frac{4x}{(x+1)(x^2+1)}=frac{A}{x+1}+frac{Bx+C}{x^2+1}.$$Bring the right-hand side to the common denominator $(x+1)(x^2+1)$. The numerators must be identically equal. It follows that$$4x=A(x^2+1)+(Bx+C)(x+1).$$Set $x=-1$. We get $-4=2A$, and therefore $A=-2$.

Đang xem: Integral 4x/(x^3+x^2+x+1)

On the right, the coefficient of $x^2$ is $-2+B$. On the left it is $0$. It follows that $B=2$.

On the right, the coefficient of $x$ is therefore $2+C$. Thus $C=2$. We conclude that$$frac{4x}{(x+1)(x^2+1)}=-frac{2}{x+1}+frac{2x+2}{x^2+1}.$$

So we want to integrate $-frac{2}{x+1}+frac{2x}{x^2+1}+frac{2}{x^2+1}$.

The rest is straightforward. To integrate $frac{2x}{x^2+1}$ let $u=x^2+1$.

Share
Cite
Follow
answered Dec 4 “13 at 4:54

*

André NicolasAndré Nicolas
483k4242 gold badges500500 silver badges924924 bronze badges
$endgroup$
Add a comment |
3
$egingroup$
The denominator has roots $i$ , $-1$ and $-i$ so you can write it in the following way $$frac{4x}{(x-i)(x+1)(x+i)}$$

Now separate it into partial fractions and integrate each of them individually .

Share
Cite
Follow
answered Dec 4 “13 at 4:41

*

abkdsabkds
2,1501414 silver badges3434 bronze badges
$endgroup$
Add a comment |

Your Answer

Thanks for contributing an answer to wnyrails.orgematics Stack Exchange!

Please be sure to answer the question. Provide details and share your research!

But avoid

Asking for help, clarification, or responding to other answers.Making statements based on opinion; back them up with references or personal experience.

See also  Integral K Over X - Integration Of Exponential Functions

Xem thêm: L Students Definition And Meaning, Law Student Definition And Meaning

Use wnyrails.orgJax to format equations. wnyrails.orgJax reference.

To learn more, see our tips on writing great answers.

Xem thêm: D Students Dedicate The Buildings ” : Nostupidquestions

Draft saved
Draft discarded

Sign up or log in

Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Submit

Post as a guest

Name
Email Required, but never shown

Post as a guest

Name
Email

Required, but never shown

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged calculus or ask your own question.

Featured on Meta
Related
2
How to integrate : $sqrt{frac{a-x}{x-b}}$
4
How to integrate $int frac{1}{sin^4(x)cos^4(x)},wnyrails.orgrm dx$?

How to integrate $frac{1}{x^4sqrt{(x^2+3)}}$

How to integrate this $frac{1}{(1-e^{2x})^{1/2}}$?
3
How to integrate $intfrac{sqrt{1+x^2}}{x},wnyrails.orgrm dx$
1
How to integrate $frac{sqrt{z+1}}{z}$
2
How to integrate this : $int frac{x^2+cos^2 x}{1+x^2}csc^2xdx$

How to integrate $int{e^{sin x}(frac{xcos^2{x}-sin{x}}{cos^2{x}}) dx}$

How to integrate this function?(substitution)
Hot Network Questions more hot questions

Question feed
Subscribe to RSS
Question feed To subscribe to this RSS feed, copy and paste this URL into your RSS reader.

*

wnyrails.orgematics
Company
Stack Exchange Network
site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. rev2021.7.23.39827

See also  Ho W = Integral Pdv Example

wnyrails.orgematics Stack Exchange works best with JavaScript enabled

*

Your privacy

By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.

See more articles in category: Integral

Leave a Reply

Back to top button