Partial Fractions Integration

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nasillmatic20

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Mar 14, 2006
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Hello everyone.... I'm stuck on a few of these rational function integrals... I don't know how to decide what A, B, and if necessary C are equal to. If somebody could help, it would be greatly appreciated!

1) ∫ dx / [x(x^2 + x -2)]

2) ∫ (x^3) / (x^2 + 2x +1)
= ∫ (x^3) / (x+1)^2

and finally...

3) ∫ dx / [x((x^2)^2)] from x=1 to x=3
 
Where are you stuck? Can you factor the denominators? Can you define the numerators?

[x(x^2 + x -2)] = x * (x+2) * (x-1)

Those are all linear terms, so...

1/[x(x^2 + x -2)] = A/x + B/(x+2) + C/(x-1)

This suggests A*(x+2)*(x-1) + B*(x)*(x-1) + C*(x)*(x+2) = 1

Now what?
 
Excellent, that's what I was having trouble with, just determining how to define ABC... so now I understand the first problem I listed, but I'm not sure about 2 and 3, especially 2, which has the numerator x^3
 
On 2, you just start with long division until it no longer has a numerator of degree 3.

You should get x - 2 + something/(x+2)^2.

Don't forget your algebra. It WILL make your life simpler.
 
I was fairly sure that I had to use long division, but I'm not certain how to do that in this case, any tips?
 
Do you know how to do it?

x<sup>3</sup> = 1*x<sup>3</sup> + 0*x<sup>2</sup> + 0*x<sup>1</sup> + 0*x<sup>0</sup>
 
nasillmatic20 said:
I honestly have no idea how to carry out the process [of polynomial long division].
Click to expand...
Since we can't really teach lessons here, please try the following online resources instead:

. . . . .Wikipedia: Polynomial Long Division

. . . . .Karl's Calculus Tutor: Polynomial Long Division

. . . . .WTAMU: Polynomial Long Division

. . . . .Polynomial Long Division

Once you have learned this technique, please re-attempt the solution method suggested earlier.

Thank you.

Eilz.
 
Another vote against teaching calculus too soon.
 
tkhunny said:
Another vote against teaching calculus too soon.
Click to expand...
I'm not aware of any information regarding this student's schedule of classes...?

It should be noted that eduators (such as the NCTM) strongly discourage the teaching of the long-division algorithm, as it is alleged to be "mere" "rote" and to "prevent the learning of deep mathematics".

So this student could have proceded through his classes on a perfectly reasonable schedule, but could easily have been prevented from previously learning this very useful technique.

Eliz.
 
[IvoryTowerSoapBox]
OK, then, another argument against failing to teach fundamental, non-sexy processes that the student will need to have well in hand before encountering calculus. In other words, "teaching calculus too soon".

I realize there is a pressing need to claim that one offers AP Courses in Mathematics. That simply cannot be done without the calculus. Nevertheless, rushing them there at the expense of things they will need makes no sense to me. Without it, they simply cannot do as well on the AP Exam. Is the point to OFFER the exam or to have the student do well on it? Where are they supposed to get it? I am certain college professors are not particularly in the mood for reviewing algebra fundamentals when they are supposed to be teaching calculus.
[/IvoryTowerSoapBox]

Interestingly, I just had a recent graduate in mathematics (B.S. from a college I won't bother to name) ask me about a simple integration problem. The student was trying to use partial fractions on a rational function with first degree numerator AND denominator. This student didn't know how to do polynomial division, either. Truly, I was stunned. There is something very wrong with this, in my view.

Note to nasillmatic20: I'm not saying anything against you. It's a systemic problem.
 
Hello everyone, thanks so much for your help... I didn't mean to start a big debate... truthbetold, I already learned polynomial division in precalc, I just totally forgot the process, as I took it back in high school. I really appreciate the help though!
 
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