Partial fractions integral

In summary, the student is trying to solve a homework equation that is not a fraction, and is having trouble getting started. The student has discovered that the problem is easier if they rewrite it into two fractions. Then they solve the fractional part.
  • #1
462chevelle
Gold Member
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9

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.
 
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  • #2
It's not partial fractions. The numerator and denominator are similar. Can you make use of that?
 
  • #3
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.
 
  • #4
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.
 

Attachments

  • calc problem.jpg
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  • #5
How would you make the numerator the same as the denominator? Something very simple.
 
  • #6
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.
 
  • #7
You're missing the obvious thing and going for more complicated things. What is the difference between the top and bottom?
 
  • #8
A constant and a sign.
 
  • #9
Do you mean 1?
 
  • #10
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.
 
  • #11
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}##
 
  • #12
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.
 
  • #13
HMM, this seems to make it easy. I would have never thought of that otherwise.
 

Attachments

  • ACalcproblem.jpg
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  • #14
I think I got it, thanks. I would have been beating my head against the wall all day otherwise.
 

Attachments

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  • #15
@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
 

1. What is a partial fractions integral?

A partial fractions integral is a mathematical technique used to decompose a rational function into simpler fractions. This allows for easier integration and is commonly used in calculus and other areas of mathematics.

2. When is a partial fractions integral used?

A partial fractions integral is used when integrating a rational function, which is a function with a polynomial in the numerator and denominator. It is also used to solve differential equations and in other areas of mathematics where integration is required.

3. How do you solve a partial fractions integral?

To solve a partial fractions integral, you first factor the denominator of the rational function into its irreducible factors. Then, you set up a system of equations using the coefficients of the original rational function and the coefficients of the decomposed fractions. Finally, you solve the system of equations to find the values of the unknown coefficients.

4. What are the different types of partial fractions integrals?

There are two main types of partial fractions integrals: proper and improper. A proper partial fractions integral is one where the degree of the numerator is less than the degree of the denominator. An improper partial fractions integral is one where the degree of the numerator is greater than or equal to the degree of the denominator.

5. Are there any special cases when solving a partial fractions integral?

Yes, there are a few special cases when solving a partial fractions integral. These include repeated linear factors, repeated quadratic factors, and irreducible quadratic factors. In these cases, the system of equations may have multiple solutions or no solutions, and additional steps may be required to find the correct coefficients.

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