Solve Integration Problems: Partial Fractions for 1/(x^2)(x-4)

In summary, the conversation is about solving a problem involving partial fractions and understanding the different forms of the solution. One user shares their solution and another user explains why it is correct. There is also a discussion about the different ways of writing the solution.
  • #1
Deathfish
86
0

Homework Statement


Someone teach me how to write equations using the editor here...

This is part of an integration question

need partial fractions of 1/(x^2)(x-4)


Homework Equations



-nil-

The Attempt at a Solution



1/(x^2)(x-4)=(Ax+B)/(x^2)+C(x-4)

Put x=0 therefore B=-1/4
put x=4 therefore C= 1/16
Put x=1 therefore A=-1/16

1/(x^2)(x-4)=(-1/16)(x+4)/x^2 +1/16(x-4)

I get a different result on Mathematica, someone help
 
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  • #2
Hi Deathfish! :smile:

If your result is

[tex]-\frac{x+4}{16x^2}+\frac{1}{16(x+4)}[/tex]

then it is completely correct!
Now, why does mathematica give another result? Well, the above result can be put in a form that is even more convenient. Indeed, the term [itex]\frac{x+4}{16x^2}[/itex] can be "simplified" further as

[tex]\frac{x+4}{16x^2}=\frac{1}{16x}+\frac{1}{4x^2}[/tex]

This is a better form since it's more suitable for integration purposes. But your result isn't wrong!
 
  • #3
The x2 factor in the denominator is not an irreducible quadratic, so there is no need for a term with Ax + b in the numerator.

I would decompose the expression in this way:
[tex]\frac{1}{x^2(x - 4)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 4}[/tex]

How you did it is essentially the same as above, since Ax/x2 is the same as A/x.
 
  • #4
Could the A/X + B/X^2 also be written as Ax+B/X^2. I think you could because splitting it up into Ax/x^2 + B/x^2 is also seen as A/x + B/x^2. Also sorry about dead post reviving...very curious.
 
  • #5
Dartx4 said:
Could the A/X + B/X^2 also be written as Ax+B/X^2.
Yes, but you need parentheses. What you wrote would be considered to be Ax + (B/x2), and I'm certain that's not what you meant.

Write it as (Ax + B)/x2.
Dartx4 said:
I think you could because splitting it up into Ax/x^2 + B/x^2 is also seen as A/x + B/x^2. Also sorry about dead post reviving...very curious.
 

What is integration?

Integration is a mathematical process of calculating the area under a curve or the accumulation of a quantity over a given interval. It is the inverse operation of differentiation and is an important concept in calculus.

What are partial fractions?

Partial fractions are a method used to simplify complex rational expressions by breaking them down into simpler fractions. This is useful in integration as it allows us to integrate each fraction separately, making the process easier.

How do I solve integration problems using partial fractions?

To solve integration problems using partial fractions, we first factor the denominator of the rational expression into linear factors. Then, we use the method of undetermined coefficients to find the coefficients of the partial fractions. Finally, we integrate each fraction separately.

What is the formula for partial fractions?

The formula for partial fractions is:
A/(x-a) + B/(x-b) + ... + N/(x-n), where A, B, ..., N are the coefficients of the partial fractions and a, b, ..., n are the linear factors of the denominator of the rational expression.

What are some tips for solving integration problems with partial fractions?

Some tips for solving integration problems with partial fractions include:
- Factor the denominator of the rational expression first
- Use the method of undetermined coefficients to find the coefficients of the partial fractions
- Check your final answer by differentiating it to see if it matches the original integrand
- Practice, practice, practice! The more problems you solve, the better you will become at using partial fractions in integration.

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