All Questions
Tagged with partial-fractions algebra-precalculus
19
questions
39
votes
4
answers
14k
views
Integration by partial fractions; how and why does it work?
Could someone take me through the steps of decomposing
$$\frac{2x^2+11x}{x^2+11x+30}$$
into partial fractions?
More generally, how does one use partial fractions to compute integrals
$$\int\frac{P(...
- 1,089
18
votes
9
answers
8k
views
Derivation of the general forms of partial fractions
I'm learning about partial fractions, and I've been told of 3 types or "forms" that they can take
(1) If the denominator of the fraction has linear factors:
$${5 \over {(x - 2)(x + 3)}} \equiv {A \...
- 7,177
25
votes
5
answers
5k
views
How does partial fraction decomposition avoid division by zero?
This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example:
$$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$
Multiplying ...
- 25.4k
6
votes
2
answers
531
views
Is this a valid partial fraction decomposition?
Write $\dfrac{4x+1}{x^2 - x - 2}$ using partial fractions.
$$ \frac{4x+1}{x^2 - x - 2} = \frac{4x+1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x-2} = \frac{A(x-2)+B(x+1)}{(x+1)(x-2)}$$
$$4x+1 = A(x-2)+B(...
- 13.6k
11
votes
6
answers
4k
views
The existence of partial fraction decompositions
I'm sure you are all familiar with partial fraction decomposition, but I seem to be having trouble understanding the way it works. If we have a fraction f(x)/[g(x)h(x)], it seems only logical that it ...
- 1,606
3
votes
4
answers
4k
views
Partial Fractions with a Repeated and a Irreducible Quadratic factor
I am trying to make this into a partial fractions form but i can't seem to find a way to do it.
The question is here:
Change into a partial fractions form.
\begin{align}
\frac{2s}{(s+1)^2(s^2 + 1)}...
- 47
1
vote
2
answers
98
views
If $S=\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\cdots+\frac{n}{1+n^2+n^4}$, then calculate $14S$.
If $$S=\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\cdots+\frac{n}{1+n^2+n^4}\,$$ find the value of $14S$.
The question can be simplified to:
Find $S=\sum\limits_{k=1}^n\,t_k$ if $t_n=\dfrac{n}{1+n^2+n^...
- 2,642
7
votes
4
answers
502
views
Why can't partial fractions expansion be "normally" done is this case?
I've learned partial fractions but I couldn't really understand one thing. When we have a case when one of the factors has multiplicity $> 1$, we got to make a kind of "stairs". e.g.
$$\frac{1}{(x-...
- 7,461
5
votes
2
answers
4k
views
Converting multiplying fractions to sum of fractions
I have the next fraction: $$\frac{1}{x^3-1}.$$
I want to convert it to sum of fractions (meaning $1/(a+b)$).
So I changed it to:
$$\frac{1}{(x-1)(x^2+x+1)}.$$
but now I dont know the next step. ...
- 271
4
votes
2
answers
230
views
Partial fractions of $\frac{-5x+19}{(x-1/2)(x+1/3)}$
Alright, I need to find the partial fractions for the expression above. I have tried writing this as $$\frac{a}{x-1/2}+\frac{b}{x+1/3}$$ but the results give me $a=25.8$ and $b=-20.8$, which are ...
- 510
2
votes
2
answers
98
views
Two partial fraction identities for $\frac{x^n}{x^m+k}$
Consider the following expression: $$\frac{x^n}{x^m+k},$$ for non-negative integers $n$ and $m$, $m>n$, and $k\in\mathbb{C}$. For $k=0$ the expression clearly simplifies to $x^{n-m}$. For $|k|>0$...
- 2,550
2
votes
5
answers
211
views
Finding sum of the series $\sum_{r=1}^{n}\frac{1}{(r)(r+d)(r+2d)(r+3d)}$
Find the sum: $$\sum_{r=1}^{n}\frac{1}{(r)(r+d)(r+2d)(r+3d)}$$
My method:
I tried to split it into partial fractions like: $\dfrac{A}{r}, \dfrac{B}{r+d}$ etc. Using this method, we have 4 equations in ...
- 671
2
votes
1
answer
431
views
How to expand $\frac{5}{(x+1)(x^2-1)}$ into partial fractions
I know how to do these problems, but this one is giving me some trouble:
Expand the following into partial fractions$$\frac{5}{(x+1)(x^2-1)}$$
Should I break it down into:
$$\frac A{x+1}+\frac B{x-1}...
2
votes
2
answers
257
views
Partial fraction decomposition of a complicated rational function
Find the partial fraction decomposition of the rational function $\displaystyle \frac{2x^3+7x+5}{(x^2+x+2)(x^2+1)}$
I have tried dividing first but keep running into problem after problem, please ...
- 2,643
2
votes
4
answers
748
views
Finding the infinite Sum of a series: $\sum\frac1{n(n+1)(n+2)}$ [duplicate]
Find the infinite Sum of the series with general term $\frac{1}{n(n+1)(n+2)}$.
I decomposed the fraction upto this $1/(2n)-1/(n+1)+1/(2n+4)$. But I find no link about cancelling terms. So how should ...
- 51