Need help with Splitting Partial Fractions involving cubic denominators (1 Viewer)

[blackops23]

Hi, I'm still coming to terms with the idea of integration by partial fractions, and there are a few questions types that I am having trouble with:

Q1. Integrate (4x^2 +1)/[(x-2)(x+2)^3]

So how do I split it?

Do i do this? -- A/(x-2) + B/(x+2) + C/(x+2)^2 + D/(x+2)^3

or do i do this -- A/(x-2) + B/(x+2) + C/(x+2)^3

Or am I wrong all together?

Help on how to split cubic factors for partial fractions would be greatly appreciated.


Also: Q2: Integrate (x^2 -1)/[(x^2)(x^2 +1)]

Once again, how do I split it?

Like this? --> A/x + B/x^2 + (ax+b)/(x^2 +1) ??

or am i just wrong?


Help on these would be greatly appreciated

Thanks guys.

 

[cutemouse]

A/(x-2) + B/(x+2) + C/(x+2)^2 + D/(x+2)^3

 

[Trebla]

There is no "wrong" way to split the partial fractions as long as it works out. The only difference between the approaches is that one ends up being easier to integrate than the other. A rule of thumb is to ensure the numerator is one degree less than the denominator.

For example, in Q2 both of the following approaches would work



They are essentially equivalent and the only difference is that when it comes to integrating the latter requires a split of the numerator in the first term anyway...

With repeated factors, you can express it in a number of ways. For the first one we can write



They are all equivalent which is easy to tell if you just combine the fractions of the latter and redefine the constants. The only difference is obviously the values of the pronumerals but they all work. The only reason the latter is preferred is because this is much easier to integrate than the others.

 

[kooliskool]

There is no "wrong" way to split the partial fractions as long as it works out. The only difference between the approaches is that one ends up being easier to integrate than the other. A rule of thumb is to ensure the numerator is one degree less than the denominator.

Erm, I disagree with what you say about any partial fraction is alright, because you can't actually let the partial fraction be "anything", the theory behind partial fraction is that you have enough variables to explain for all the degree in the denominator, no less and no more.

For example if the denominator is degree four, you at least need four coefficients.

That was the only thing I want to clarify, hope that helps

P.S. For partial fraction to work, the fraction to be equating with has to be proper fraction, that means, the degree of numerator has to be at least one degree less than degree of denominator, if it isn't, then that's why you do long division in polynomials.

 

[funnytomato]

There is no "wrong" way to split the partial fractions as long as it works out ... A rule of thumb is to ensure the numerator is one degree less than the denominator.

≈

...you can't actually let the partial fraction be "anything"

... that means, the degree of numerator has to be at least one degree less than degree of denominator

 

[Trebla]

Erm, I disagree with what you say about any partial fraction is alright, because you can't actually let the partial fraction be "anything", the theory behind partial fraction is that you have enough variables to explain for all the degree in the denominator, no less and no more.

For example if the denominator is degree four, you at least need four coefficients.

That was the only thing I want to clarify, hope that helps

P.S. For partial fraction to work, the fraction to be equating with has to be proper fraction, that means, the degree of numerator has to be at least one degree less than degree of denominator, if it isn't, then that's why you do long division in polynomials.

I didn't explicitly say the partial fraction could be 'anything' (hence the use of "as long as it works out") though I could probably see why it may have been implied. Just to clarify, the point I was trying to make was that the multiple forms suggested above in the original post are all correct (none of the them are "wrong"). The only reason one form is preferred over the other is because the integration steps become easier with this particular form.
For example, in the case of multiple factors (e.g. a double factor), it doesn't matter if you use
a/(x + 2) + b/(x + 2)² or (ax + b)/(x + 2)²
but obviously the first is easier to integrate than the second...

 

[kooliskool]

That's kool, I was just trying to clarify too.

 

[blackops23]

Can someone provide a solution on how to solve (4x^2 + 1)/((x-2)(x+2)^3)? I can't seem to do it

Thank you

 

[AAEldar]

Can someone provide a solution on how to solve (4x^2 + 1)/((x-2)(x+2)^3)? I can't seem to do it

Thank you

Then solve for A, B, C and D. I found it easier to sub values for x (I used -2, 0, 1 and 2), then to equate co-efficients.

 

[cutemouse]

NB: Repeated factors for partial fractions are not in the HSC course. Although they can ask something similar. eg. giving you the form of the partial fractions and asking you to find the unknown coeffs.