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#1 2014-08-19 15:05:52

Au101
Member
Registered: 2010-12-01
Posts: 353

Difficult partial fractions and/or difficult integral

Hi again, so the next question that has me completely stumped is:

I have completed the proof, but am stuck on the next half of the question. Because of what it asked me to prove, I attempted to solve the second integral and my method was to try using the substitution:

Here's what I have:

And here is where I get completely stuck. WolframAlpha tells me that this does have a partial fraction expansion, but I can't for the life of me work out how to arrive at it, since neither:

Nor:

Factorise, meaning that I cannot make either of them zero.

So, either I'm missing a trick and there is something I can do to make these zero, or there's a partial fractions technique I've not learned, or I'm barking up entirely the wrong tree with my substitution.

Any ideas?

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#2 2014-08-19 18:00:54

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Difficult partial fractions and/or difficult integral

Hi;

Try this here:

now substitute u = 0 , 1, -1, 2 and solve the 4 x 4 simultaneous set of linear equations.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2014-08-19 19:06:53

Bob
Administrator
Registered: 2010-06-20
Posts: 10,059

Re: Difficult partial fractions and/or difficult integral

hi Au101,

What's wrong with

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#4 2014-08-20 01:11:33

Au101
Member
Registered: 2010-12-01
Posts: 353

Re: Difficult partial fractions and/or difficult integral

bob bundy wrote:

hi Au101,

What's wrong with

Bob

I never really knew how to do that, but thanks bob bundy, that way's much easier.

Still, I'm glad I now know how to do the partial fractions - thanks bobbym smile

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#5 2014-08-20 01:41:11

Au101
Member
Registered: 2010-12-01
Posts: 353

Re: Difficult partial fractions and/or difficult integral

What was your method for solving the four simultaneous equations bobbym? I have now done it, but it took me multiple pages!!

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#6 2014-08-20 01:44:07

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Difficult partial fractions and/or difficult integral

Hi;

That is best done by a computer but it probably can be done by hand too.

1) Clear out fractions and use the 2nd equation to reduce to a 3 x 3.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2014-08-20 01:55:37

Au101
Member
Registered: 2010-12-01
Posts: 353

Re: Difficult partial fractions and/or difficult integral

Ta! smile I can show you how I did it, for the sake of interest, but it's nice to know there wasn't a simple method I completely missed:

To be continued...

Last edited by Au101 (2014-08-20 01:59:29)

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#8 2014-08-20 01:58:49

Au101
Member
Registered: 2010-12-01
Posts: 353

Re: Difficult partial fractions and/or difficult integral

...

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#9 2014-08-20 02:02:18

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Difficult partial fractions and/or difficult integral

Hi;

We can clean up the original equations to be:




Use b = 1 - d and substitute:



Now use a = -c - 1 to reduce to a 2 x 2


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#10 2014-08-20 02:05:03

Au101
Member
Registered: 2010-12-01
Posts: 353

Re: Difficult partial fractions and/or difficult integral

Ah yes, that would have worked very nicely. Much neater!

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#11 2014-08-20 02:06:51

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Difficult partial fractions and/or difficult integral

We will get:

You can now use c = d - 1 to reduce to a linear equation.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#12 2014-08-20 02:19:14

Au101
Member
Registered: 2010-12-01
Posts: 353

Re: Difficult partial fractions and/or difficult integral

Certainly, thanks a lot smile

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#13 2014-08-20 02:20:43

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Difficult partial fractions and/or difficult integral

I made a little change to the last post. After you have d, you back substitute into the equations above it and everything will be fine.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#14 2014-08-20 04:15:27

Bob
Administrator
Registered: 2010-06-20
Posts: 10,059

Re: Difficult partial fractions and/or difficult integral

Au101 wrote:

I never really knew how to do that, but thanks bob bundy, that way's much easier.

I just thought, why did they ask for that substitution, unless it made the integral easier.  So in what way ? Thinks .......  Ah ha! 

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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