View Full Version : Integral help
AtlasShrugs
07-23-2008, 09:25 PM
Integrate 1/(x^4-1), I know that I must use partial fractions and that the starting fraction is split up into (x-1), (x+1), and (x^2+1). However I am terrible with partial fractions, so could someone show me how to do the rest of this problem.
HallsofIvy
07-25-2008, 11:09 AM
Integrate 1/(x^4-1), I know that I must use partial fractions and that the starting fraction is split up into (x-1), (x+1), and (x^2+1). However I am terrible with partial fractions, so could someone show me how to do the rest of this problem.
So you know that you can write
1/(x^4-1)= A/(x-1)+ B/(x+1)+ (Cx+ D)/(x^2+ 1)
Multiplying both sides by x^4- 1 we can cancel factors on the right and get
1= A(x+1)(x^2+ 1)+ B(x-1)(x^2+1)+ (Cx+ D)(x-1)(x+1).
There are 4 unknowns so we need four equations. Since this must be true for all x, we can choose any 4 values of x to get those 4 equations (we could also multiply out all the terms on the right and "equate like coefficients" but that is tedious.)
Seeing the factors (x- 1) and (x+ 1) should tell us that choosing x= 1 and x= -1 (which make those equal to 0) will be especially easy:
If x= 1, then we have 1= A(1+1)(1+1)+ B(1-1)(1+1)+ (C+ D)(1-1)(1+1) and only the first term on the right is non-zero: 1= 4A so A= 1/4.
If x=-1, then we have 1= A((-1+1)(1+1)+ B(-1-1)(1+1)+ (-C+D)(1+1)(-1+1) and now only the second term is non-zero. 1= -4B so B= -1/4.
Unfortunately, there is no real number such that x^2+ 1= 0 so the last two, C and D, are not that easy. But we can still just choose 2 values for x and it is probably best to choose x to be the simplest numbers left: 0 and 2 since we have already used 1 and -1.
If x= 0, then 1= A(0+1)(0+1)+ B(0-1)(0+1)+ (0+D)(0-1)(0+1)= A- B- D. And we already know that A= 1/4 and B= -1/4 so A+ B= 0. D= -1.
If x= 2, then 1= A(2+1)(4+1)+ B(2-1)(4+1)+ (2C+ D)(2-1)(2+1)= 14A+ 5B+ (2C+ D)(3). Since we already know that A= 1/4, B= -1/4, and D= 1, that tells us that 15/4- 5/4+ 6C+ 3= 1 or 6C= 1- 3- 10/4= 1- 3- 5/2= -9/2. C= (-9/2)/6= -3/4.
1/(x^4- 1)= (1/4)/(x-1)- (1/4)/(x+1)- (x+ 3/4)/(x^2+ 1).
Nelykur
02-28-2010, 12:47 PM
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ADVANCED MODE - PSG1
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5--:--:-------
Level 10
100:58:63Psycho Mantis
201:11:27Raven
3--:--:-------
4--:--:-------
5--:--:-------
Level 11
100:26:65Raven
200:27:80Psycho Mantis
3--:--:-------
4--:--:-------
5--:--:-------
SPECIAL MODE - VR Mission
102:31:39Psycho Mantis
202:43:41Raven
305:46:54Gumi
406:42:96Sayk
5--:--:---------
SPECIAL MODE - Ninja
Level 01
100:29:41Psycho Mantis
200:39:51Raven
300:41:43Gumi
400:42:30Jojek
500:53:70Saladin
Level 02
101:01:29Psycho Mantis
201:22:83Gumi
301:23:22Raven
401:50:82Sayk
502:40:80Jojek
Level 03
100:09:31Psycho Mantis
200:09:54Gumi
300:10:13Raven
400:10:43Sayk
5--:--:---------
SPECIAL MODE - NG Selection
Level 01
100:07:48Psycho Mantis
200:09:38Raven
300:10:12Sayk
4--:--:-------
5
Level 02
100:04:42Psycho Mantis
200:04:42Raven
300:04:43Sayk
4--:--:-------
5--:--:-------
Level 03
100:15:60Psycho Mantis
200:17:29Raven
300:17:41Sayk
4--:--:-------
5--:--:-------
Level 04
100:05:65Psycho Mantis
200:05:67Raven
300:07:60Sayk
4--:--:-------
5--:--:-------
Level 05
100:14:52Psycho Mantis
200:19:56Raven
3--:--:-------
4--:--:-------
5--:--:-------
Level 06
100:03:65Psycho Mantis
200:06:66Raven
3--:--:-------
4--:--:-------
5--:--:-------
Level 07
100:17:65Psycho Mantis
200:18:06Raven
3--:--:-------
4--:--:-------
5--:--:-------
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