Difficult integral problem

1. Nov 6, 2014

Austin

1. The problem statement, all variables and given/known data
Indefinite integral of 1/(x(x+1)(x+2)...(x+m))dx

2. Relevant equations
None

3. The attempt at a solution
The problem seems to be a conceptual one and naturally in a format like that I think partial fractions expansion... However that does not seem to work here.. Thanks for any help I am lost on where to start

2. Nov 6, 2014

vela

Staff Emeritus
Why doesn't partial fractions work?

3. Nov 6, 2014

Austin

I thought since there are infinite terms in the denominator it would never happen?

4. Nov 6, 2014

vela

Staff Emeritus
The way you wrote it, it appears to be a finite number of factors in the denominator. Is it supposed to be infinite?

5. Nov 6, 2014

Austin

I may not have full understanding of the question because when I saw the "..." Followed by the x + arbitrary m, i assumed that meant infinite or atleast undefined. Do you see it differently?

6. Nov 6, 2014

Austin

The question is written in the textbook the way I wrote it in the question just to clarify, with the "..."

7. Nov 6, 2014

vela

Staff Emeritus
That would mean there are m+1 factors: x, x+1, ..., x+(m-1), and x+m. The result will depend on m, obviously. It's like when one writes $n! = 1\times2\times3\times\cdots\times n$. Just because a specific value for $n$ isn't given, you don't assume there's an infinite number of factors.

Please note that I'm not saying partial fractions is the best way to do this problem, but without knowing what you've been studying, I think it's a reasonable approach.

8. Nov 6, 2014

Austin

I am in calc bc this problem is supposed to be a difficult college level problem... I understand what you mean that it doesnt mean infinity but it is undefined so i cant see how i could apply partial fractions to this??

9. Nov 6, 2014

Austin

And if partial fractions wont work then what can I do??

10. Nov 6, 2014

vela

Staff Emeritus
I wouldn't say $m$ is undefined. You can assume it's a positive integer. It might help to try work out the partial fraction expansion for a specific value of $m$, like m=5 for example, and see if a pattern emerges that you can generalize for arbitrary $m$. Finally, I suggest you use the Heaviside cover-up method to solve for the various constants.

EDIT: Here's the expansion for $m=5$ from Mathematica. It does appear there's some sort of pattern. Can you figure out what it is?
$$\frac{1}{120 x}-\frac{1}{24 (x+1)}+\frac{1}{12 (x+2)}-\frac{1}{12 (x+3)}+\frac{1}{24 (x+4)}-\frac{1}{120 (x+5)}$$

11. Nov 6, 2014

Ray Vickson

For finite m, partial fractions will work, but they won't be particularly pretty. Assume
$$f(x) \equiv \frac{1}{x(x+1)(x+2) \cdots (x+m)} = \frac{A_0}{x} + \frac{A_1}{x+1} + \frac{A_2}{x+2} + \cdots + \frac{A_m}{x+m}$$
Then
$$x f(x) = \frac{1}{(x+1) \cdots (x+m)} = A_0 + \sum_{i=1}^m \frac{A_i x}{x+i}$$
so setting $x = 0$ gives $1/m! = A_0$. Similarly,
$$(x+1) f(x) = \frac{1}{x(x+2) \cdots (x+m)} = A_1 + \sum_{i \neq 1}\frac{ A_i (x+1)}{x+i}$$
so setting $x = -1$ gives
$$A_1 = \frac{1}{(-1)(-1+2) \cdots (-1+m)} = -\frac{1}{(m-1)!}$$
Continue like that to get all the $A_i$.

12. Nov 6, 2014

Austin

I feel like this explanation is good however I have not had much experience with summations so it is difficult for me to fully understand your explanation. I am beginning to see the pattern but I am not sure how I would express this so I would be able to take the integral. I am trying very hard to understand what you are saying but I just haven't had much experience with problems like this so I am having trouble putting it all together.

13. Nov 6, 2014

Austin

more specifically I am confused about the equation you set up and the process you are using to find the A values

14. Nov 6, 2014

Austin

Additionally, since I cannot tell if m is an odd or even integer how would I tell which of the pieces of the partial fraction will be negative?

15. Nov 6, 2014

Austin

So far I believe my partial fractions will look something like this:

1/((m!)(x))- 1/(((m-1)!))(x+1)) +1/(((m-2)!)(x+2)) +...+1/(((m-2)!)(x+(m-2))) - 1/(((m-1)!)(x+(m-1))) + 1/((m!)(x+m))

I am not sure if this is getting anywhere?

Sorry for it looking messy I am not sure how to write fractions on this website yet if anyone could tell me?

16. Nov 6, 2014

vela

Staff Emeritus
Did you try working it out for m=5?

17. Nov 6, 2014

Austin

I guess it would not work then because for the 1/x+2 term it should be 12 but with my model i got 6 but i really dont know what to do from here

18. Nov 6, 2014

Ray Vickson

You need to work it out systematically, the way I illustrated for the first two terms 1/x and 1/(x+1). Also, trying it out systematically from start to finish for m = 4 and m = 5 will go a long way towards helping you understand the method and result.

19. Nov 7, 2014

Ray Vickson

It is even more revealing to re-write this after dividing and multiplying by 5! = 120:
$$f(x) = \frac{1}{5!} \left( \frac{1}{x} - \frac{5}{x+1} + \frac{10}{x+2} - \frac{10}{x+3} + \frac{5}{x+4} - \frac{1}{x+5} \right)$$
The successive coefficients inside ( ) ought to be instantly recognizable.