How Do You Integrate (x^2+x)/(x^2+x+1) from 0 to 1?

[462chevelle]

Homework Statement

integral(0>1) of (x^2+x)/(x^2+x+1)dx

Homework Equations

Factor denominator, and set numerator with A,B,C, etc. multiply both sides by the common denominator.

The Attempt at a Solution

Since the denominator won't factor at all I don't really know where to start, I could rewrite it into 2 fractions or factor the numerator. Both useless though, I can't find an example in my book for this type of problem.

Any hints on the first step?
Thanks.

 

[PeroK]

It's not partial fractions. The numerator and denominator are similar. Can you make use of that?

 

[462chevelle]

Almost, as far as I can tell, if I take the denominator to be u I get du/(2x+1)=dx then I can factor a x out of the numerator and i get x(x-1)/(u(2x+1)). Unless I'm missing or forgetting something.

 

[462chevelle]

Sorry, I'm bad at latex so I'm posting a picture of my work. I can't remember if that is allowed or not.

 

[PeroK]

How would you make the numerator the same as the denominator? Something very simple.

 

[462chevelle]

You could add 2x+1 and that would be the derivative of the denominator. Then I would have to do that to the denominator. I'll see what I can figure out and post my work.

 

[PeroK]

You're missing the obvious thing and going for more complicated things. What is the difference between the top and bottom?

 

[462chevelle]

A constant and a sign.

 

[PeroK]

Do you mean 1?

 

[462chevelle]

Well, yes the constant would be 1. I just realized I made a typo on the first post, sorry. Its (x^2-x)/(x^2+x+1) the problem is correct in the picture I posted.

 

[PeroK]

That makes the solution a lot easier you'll be glad to know!

Let me do the original one and see if that helps:

$\frac{x^2 + x}{x^2+x+1} = 1 - \frac{1}{x^2+x+1}$

 

[HallsofIvy]

Since numerator and denominator have the same degree, first do the division to get a polynomial plus a fraction with numerator of lower degree than the denominator. Then complete the square in the denominator.

 

[462chevelle]

HMM, this seems to make it easy. I would have never thought of that otherwise.

 

[462chevelle]

I think I got it, thanks. I would have been beating my head against the wall all day otherwise.

 

[Mark44]

@462chevelle, since we saved you a whole day of beating your head against the wall, maybe you could devote some of that saved time to learning a bit of LaTeX? It's really not very hard. We have a brief summary here: https://www.physicsforums.com/help/latexhelp/

Here are a few of the things you could have used in your problem. Note that I have omitted the # # pairs (without the space) at the beginning and end in my examples below. I did that so that the LaTeX script remains visible.

Exponents:
x^2
e^{x + 1}

Fractions:
\frac{x^2 + x}{x^2 + x + 1}

Integrals:
\int 3x^2 dx

\int_0^3 e^{x + 1} dx