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Old 09-03-2004, 02:48 PM   #1 (permalink)
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Calculus 2 is the best calculus ever!!!

As much as i hate asking people for help, i'm stumped. The problem involves integration by partial fractions. If you want to help me the most just give me a hint, don't solve the fucker for me.

Anyways the problem is to integrate the function 1/(x^4+1) by way of adding and then subtracting from (x^4+1) to create a difference of squares.

My first idea was to subtract two from the denominator resulting in

1/[(x^4-1)+2]

which gets factored moreso down to

1/[(x^2+1)(x+1)(x-1)+2]

normally you'd want to make a substitution, but with all of the factors down there that doesn't seem to make much sense.

That's where i'm stuck.
Maybe i should've made the substitution
u=(x^4-1) giving the
du=4x^3dx
x=(u-1)^(1/4)

That would give me the integrand
1/[4(u-1)^(3/4)*(u+2)]

Anyways, i've got to work. If you think i'm on the right track let me know. Thanks in advance.
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Old 09-03-2004, 04:03 PM   #2 (permalink)
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Oh, scheisse. I'm gonna have to do that soon (starting AP Calc). Um... good luck to both of us.
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Old 09-03-2004, 04:13 PM   #3 (permalink)
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It's like trying to run up a muddy hill in the pouring rain. Doable, but a pain in the ass nonetheless.
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Old 09-03-2004, 11:25 PM   #4 (permalink)
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That problem is resonating my brain from my calc 2 class last spring, but let me wait till morning to see if I can post anything helpful, I'm about dead from exhaustion.
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Old 09-04-2004, 06:37 AM   #5 (permalink)
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Try this:

Factorize (x^4+1) like this:

Code:
(x^4+1) = (x^2+ax+b)(x^2+cx+d)
So you'll have:

Code:
1/(x^4+1) = 1/(x^2+ax+b)(x^2+cx+d)
Then, try to write

Code:
1/(x^2+ax+b)(x^2+cx+d)
like this:

Code:
P(x)/(x^2+ax+b) + Q(x)/(x^2+cx+d),
where P(x) and Q(x) are polynomials of first degree or lower.
Then you'll be able to integrate each part of that sum separately. You only need a simple substitution for each one to do it.

Last edited by agve; 09-04-2004 at 06:39 AM..
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Old 09-04-2004, 06:41 AM   #6 (permalink)
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I have a degree in Applied Mathematics and therefor have taken enough Calc and Diff-E-Q to choke a horse.

But that was over 18 years ago and these days I don't fumble with numbers enough to even balance my checkbook!

That equation sent chills up my spine. I'm no racist, but I always hated integration.

I did find THIS. It's close, I think. And maybe even helpful.

Good luck.
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Old 09-04-2004, 09:56 AM   #7 (permalink)
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Thanks for the suggestions. I was just about to have another crack at it. I'm going to try the agve method, but i'm kind've unclear on how to find the value of the coefficients a, b, c, and d. Is there any kind of systematic method or is it all guess and check. Platypus, that link will be helpful for other integrals, but it doesn't have the one that i need, thanks though.
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Old 09-04-2004, 11:36 AM   #8 (permalink)
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I found those coefficients a while back, and I know them, but I forgot how I did it. I'll post you a message when I remember it.

I could also tell you the coefficients, but I don't think that's what you want.

Edit:

Have you solved it yet? I still can't remember exactly what I did. I found them by playing with some related expressions involving i...

Last edited by agve; 09-06-2004 at 06:57 PM..
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Old 09-07-2004, 01:15 PM   #9 (permalink)
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Actually, about two pages later and a quick reference to some cryptic old english-like calculus text some professor posted on the internet, i think i'm done. The actual problem took a page, which is something i guess i should get used to. You were right in your suggestion agve. (x^4+1) = (x^2 + sqrt(2)x +1)(x^2 - sqrt(2)x + 1) I still didn't end up with the same answer that was in the text that i found, but it's damn close and my writing hand is too cramped up right now for me to want to mess with it anymore. Either this problem is ridiculously difficult for a student at my progress level or i have gotten dumber in the last few days. Probably a mixture of the two, though the text i found said leibnitz himself found this integration problem "troubling". Anyways, thanks for the suggestions.

Last edited by filtherton; 09-07-2004 at 01:17 PM..
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Old 09-08-2004, 04:56 PM   #10 (permalink)
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That was the one thing I hated about Calc 2 - It made me feel so dumb, and I'd like to think I was a pretty bright student.
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Old 09-09-2004, 09:54 AM   #11 (permalink)
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Calc sucks.

I took Calc 1, 2, and 3 from Comm. College in town in high school before going off to college. Let me say, never yet in my college experience have I found something as dry AND difficult at the same time. After those classes, I haven't even touched anything algebraic. It scares me now to think I even did it! Only 2 years ago!!
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Old 09-09-2004, 12:13 PM   #12 (permalink)
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Calc is actually more interesting and meaningful than algebra ever was. I love it.

I ALWAYS wondered where shit like "Volume of Sphere = (4*pi*r^3)/3" came from.. I always asked "WHY is it that?" and the teachers always said "because it is".

Now I know why it is the way it is! Past teachers could've at least told me "you'll find out in calc"

It's actually pretty interesting... and I'd try your problem out, but we haven't done the method you're being taught yet. We're doing integration by parts with the whole "u dv = uv - (u'v dx)" method.

[edit]
Thinkin out loud here.. but would arctan work? (nevermind, forgot you were doing partial fractions)
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Last edited by Stompy; 09-09-2004 at 12:27 PM..
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Old 09-09-2004, 12:47 PM   #13 (permalink)
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Location: In the land of ice and snow.
Quote:
Originally Posted by Stompy
Calc is actually more interesting and meaningful than algebra ever was. I love it.

I ALWAYS wondered where shit like "Volume of Sphere = (4*pi*r^3)/3" came from.. I always asked "WHY is it that?" and the teachers always said "because it is".

Now I know why it is the way it is! Past teachers could've at least told me "you'll find out in calc"

It's actually pretty interesting... and I'd try your problem out, but we haven't done the method you're being taught yet. We're doing integration by parts with the whole "u dv = uv - (u'v dx)" method.

[edit]
Thinkin out loud here.. but would arctan work? (nevermind, forgot you were doing partial fractions)

I agree. Whilst really obnoxious at times, i have to admit that i like the calulus. It's so clever. If your book is anything like mine then you're almost at the point where you could to that problem. We covered substitution by parts a few sections before this problem.

If you're interested in a summary of the solution and the problem it is located on this page in section 4.5.5 at the bottom. Arctan is part of it, but there are also a few ln's in the mix too.
http://kr.cs.ait.ac.th/~radok/math/mat6/calc4.htm
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Old 09-13-2004, 01:13 PM   #14 (permalink)
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I learned it as uv - (vdu) too, filtherton, satisfy my curiosity, is the way your talking about a different subject and you learned the UV thing earlier, or is it just another way of doing the same problem?
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Old 09-15-2004, 07:37 AM   #15 (permalink)
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In the class i'm taking we learned the udv - integral(vdu). We learned it a few sections before this problem was foisted upon us. I think it is just another way to do the problem.
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