Question

Integration question below:

Answer 1

\[\int\limits_{ }^{ }\frac{ x }{ (x+1)^2(x^2 +1) }dx\]

Answer 2

Now, when I split this question into partial fractions to integrate, I get \[\frac{ A }{ x + 1 } + \frac{ B }{ (x+1)^2 } + \frac{ Cx + D }{ x^2 + 1 }\] while the memo gives \[\frac{ A }{ (x+1)^2 } + \frac{ Bx + C }{ (x^2 + 1) }\]. What rule am I not taking into account?

Answer 3

that is weird... I would have split up the x^2+1 too though

Answer 4

You are doing it right.

Answer 5

maybe that modifies somewhere, let me write it out

Answer 6

First one looks good to me..

Answer 7

Just find out the constants, you will get the same.

Answer 8

Scratch that it's a plus can't use conjugates

Answer 9

ya your setup looks good Cog :)
You have a repeated linear factor, so you have to split up the A and B like that.

Answer 10

Just in case you want a secondary reference, but I agree looks good: http://www.purplemath.com/modules/partfrac2.htm

Answer 11

is that a rule when you have square factors in the denominator ?

Answer 12

no, the memo is incorrect. You did it by the rules.

Answer 13

You may just end up with A=0 though

Answer 14

@rsadhvika yes, when you got square or cubes, then you do like that.

Answer 15

this looks a lot like \[\large 0.\overline{123} = \frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \cdots\]

Answer 16

It is surely the case, if first constant will be \(0\), then first term will vanish, and you are left with second and third only.

Answer 17

But, @incognito You should still set it up as you did

Answer 18

i remember reading it somewhere but not really sure how exactly they are related

Answer 19

http://www.purplemath.com/modules/partfrac2.htm @rsadhvika , you can read all of the rules here

Answer 20

I think they are related in your eyes only. :P

Answer 21

it doesn't explain "why" that rule works

Answer 22

Are partial fractions rule?? Or they are just conventions?

Answer 23

I don't have a good understanding of why it works myself.
There was a really interesting explanation on stackexchange, maybe check it out
http://math.stackexchange.com/questions/20963/integration-by-partial-fractions-how-and-why-does-it-work

Answer 24

exactly what i am looking for, thanks @zepdrix !

Answer 25

I would say they are a form of factoring. Essentially what you do is break down the bottom into pieces then facilitate the addition which gives you your bottom back, and then you have to solve the system with the original

Answer 26

Okay, thanks, I've gotten two of the same coefficients as given in the memo, those for B and C in my expression, but I can't find A and D? all I get is that A + D = 1/2.

Answer 27

if you type out the intensive FOILing, I'll check it

Answer 28

I rearranged my expression to look like \[\frac{ D }{ x+1 }+\frac{ A }{ (x+1)^2 }+\frac{ Bx + C }{ (x^2 + 1) } = x\] so that the coefficients I am looking for will correspond with the memo, and then I get the expression \[D(x+1)(x^2+1) + A(x^2+1) + (Bx + C)(x+1)^2 = x\] into which I substituted values of x, namely -1, 0, and 1. That's how I've been doing it.

Answer 29

From which I got, A = -1/2 and B = 0, and C + D = 1/2.

Answer 30

@lncognlto please check my method

Answer 31

my b which is your A is also -1/2 and My D which is yourC is 1/2, the other two were zero, but instead of inputting x values, you should just solve the system of eqs

Answer 32

@FibonacciChick666 please check my attachment

Answer 33

thats a clever method!

Answer 34

That is just a rearrangement, but good use of brain I must say. :)

Answer 35

It reduces to standard forms.:). we dont ahve to use partial fractions.:)

Answer 36

nice rearrangement

Answer 37

Okay, I got it. Thanks guys