Rewriting rational function in the form of partial fractions is often useful when calculating integrals.

learn more… | top users | synonyms

4
votes
4answers
62 views

Partial fractions and trig functions

A long time ago I wrote down a silly problem. It starts with Attempt to write $$\frac{1}{\sin(x)\cos(x)}$$ using partial fractions. and then goes on to prove a trig identity. I was wondering if ...
0
votes
1answer
40 views

Differential Equation with biology!

I am working on a growth model for bacteria as a function of a nutrient, and I am stuck. So the differential equation I am supposed to be solving is $\frac{dN}{\ DT} = k(C_0 -\alpha N(T)) N$ The ...
0
votes
1answer
28 views

How can I perform Partial Fractions Decomposition on a Telescoping Series involving Exponentials?

Given: $\sum_{k = 1}^\infty\dfrac{6^k}{(3^k - 2^k)(3^{k+1} - 2^{k+1})}$ The Partial Fractions Decomposition is: $\sum_{k = 1}^\infty(\dfrac{3^k}{3^k - 2^k} - \dfrac{3^{k+1}}{3^{k+1} - 2^{k+1}})$ ...
0
votes
0answers
36 views

Example: How to find inverse Laplace Transform by integral of the function (5.2-29)

This is just a demonstration on how to solve the following type of problem. Find $\mathcal{L}^{-1}\{\frac{54}{s^3(s-3)}\}$ by the given method: $$\mathcal{L}\{ ...
0
votes
0answers
23 views

Expanding a partial fraction…

$\dfrac1{x^2+3}=x^{-2}\left(1+\dfrac{3}{x^2}\right)^{-1} = x^{-2}\left(1-\dfrac{3}{x^2}+\dfrac{9}{x^4}+\cdots\right)$ But this can also be factored into: ...
0
votes
0answers
42 views

partial fraction decomposition

$$\frac{x^6−x^5+48x^3−53x^2+99x+48}{x^5−2x^4+2x^3−4x^2+x−2}$$ Initially I used long division and then used trial and error to factor the denominator and ...
-1
votes
1answer
23 views

. Find the partial fraction decomposition of the following rational function.

Find the PFD of the folowing: $$\frac{x^6-x^5+48x^3-53x^2+99x+48}{x^5-2x^4+2x^3-4x^2+x-2}$$ Initially I used long division and got $$x+1+\frac{50(x^3-x^2+2x+1)}{x^5-2x^4+2x^3-4x^2+x-2}$$ I then used ...
-2
votes
1answer
49 views

Calculate $S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}$. [duplicate]

Calculate $S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}$. I know I posted this question already but I want a more detailed answer. For example, how you got from one step to another using the partial fraction ...
1
vote
3answers
93 views

Indefinite integration problem $\int {\frac{1}{1+ \tan^4 x}}dx$

The following problem came up in my last examination. $$ \int {\frac{1}{1+ \tan^4 x} dx}$$ The difficulty I was facing was that I wasn't able to find anything to substitute as there was nothing ...
1
vote
1answer
41 views

Why does Partial Fractions Decomposition fail for higher degree nominator?

I can decompose $${1\over(x-a)(x-b)} = {1\over(a-b)}({1\over(x-a)}-{1\over(x-b)}) = {1\over(a-b)}({x-b-x+a\over(x-a)(x-b)}) = {a-b \over (a-b)(x-a)(x-b)}$$ and $${x\over(x-a)(x-b)} = ...
2
votes
2answers
38 views

Decomposing $\frac{(\omega-1)^2}{(1+\omega^2)^2}$ into partial fractions

How can I decompose $$\frac{(\omega-1)^2}{(1+\omega^2)^2}$$ into partial fractions? Should I solve $$\frac{A\omega + B}{1+\omega^2} + \frac{C\omega^3 + D\omega^2 + E\omega + D}{(1+\omega^2)^2} = ...
1
vote
3answers
40 views

Do I need to use partial fractions to find $\frac{2n+1}{n^2(n+1)^2} = \frac{1}{n^2}-\frac{1}{(n+1)^2}$?

I need to simplify $\frac{2n+1}{n^2(n+1^2)}$ as part of an exam question. The solution states $$\frac{2n+1}{n^2(n+1)^2} = \frac{1}{n^2}-\frac{1}{(n+1)^2}$$ In the solution it does not state how this ...
1
vote
3answers
36 views

integration using partial fraction with repeating denominator

So i need to integrate this $1/[(x^3)(x-1)]$, that means it could be decomposed into: $$A/x + B/x^2 + C/x^3 + D(x-1)$$ Furthermore the resulting equation would then be: $1 = A(x^2)(x-1) + Bx(x-1) ...
0
votes
1answer
60 views

Finding $\int \frac{1+\sqrt{\frac{x+1}{x-1}}}{1-\left(?\frac{x+1}{x-1}\right)^{\frac14}}dx$ part2

I am finishing computing Finding $\int \frac{1+\sqrt{\frac{x+1}{x-1}}}{1-\left(\frac{x+1}{x-1}\right)^{\frac14}}dx$. . I have question regarding partial fractions. What would be the efficient way to ...
6
votes
1answer
56 views

An identity involving partial fractions decompositions

In Vladimir A. Smirnov's book Analytic Tools for Feynman Integrals (page 38), the following identity is suggested to perform partial fractions decompositions $$ \begin{split} ...
2
votes
1answer
32 views

Multi Variable Partial Fraction Problem

The question is to develop $$(x-\alpha)(x-\beta)(x-\gamma)\over(x-a)(x-b)$$ into partial fractions. Someone challenged me to solve this question and said the answer is ...
2
votes
3answers
33 views

Explanation solution partial-fraction of $\frac{x^2 + 2}{x^2 - 1}$

The partial fraction of $\dfrac{x^2+2}{x^2-1}$ is $1 + \dfrac{3}{2}\cdot(\dfrac{1}{x-1}-\dfrac{1}{x+1})$. I understand how you get $\dfrac{3}{2}\cdot(\dfrac{1}{x-1}-\dfrac{1}{x+1})$ but from where ...
2
votes
1answer
53 views

Partial Fraction Decomposition of Exponential Generating Functions

I want to see if it is possible to write $$ \left(\frac{x}{e^x-1}\right) \left(\frac{x^2/2! }{e^x-1-x}\right) \left(\frac{x^3/3!}{e^x-1-x-x^2/2}\right)$$ as a linear combination of the factors ...
1
vote
0answers
35 views

Why do repeated linear factors have to be dealt with in this way?

When dealing with partial fractions, and your denominator has a repeated linear factor, the way to solve is this: $\frac{2x+3}{(x-2)^2}=\frac{A}{(x-2)^2}+\frac{B}{(x-2)}$ $2x+3=A+B(x-2)$ and so on. ...
1
vote
1answer
33 views

Recurrence relation stuck on partial fraction decomposition

I am stuck in trying to solve the following recurrence relation: $$S_n = rS_{n-1} + nrB$$ where $r$ and $B$ are constants. To solve this I made the following generating function: $$f(x) = ...
0
votes
0answers
37 views

Tricky partial fraction decomposition

Probably not the best title, but I don't have any other ideas. I have to find the inverse Z-transform of this fraction: $\frac{3 z^6-10 z^5+6 z^4+10 z^3-6 z^2+3z}{z^7-3 z^6+2 z^5+36 z^2-51 z+15}$ ...
0
votes
1answer
29 views

Combining two fractions involing powers of x

Is there any way i can write $x^a+x^b$ as d$x^c$ Im considering writing letting $a=a-1$ and partial fractions but im getting really confused.
1
vote
1answer
38 views

Partial Fractions Variables

$$\int\frac{x^3+6x^2+3x+16}{x^3+4x}\,dx$$ Eventually one solves that a variable (I used $C$) ${}= -x$. By variables I mean the decomposition yields $A + Bx+C$. Therefore $C = -1$. I think that $C$ ...
0
votes
2answers
41 views

How to re-write one fraction as two others.

I have the two following fractions. $$ \dfrac{A}{Bx^{\alpha+1}}$$ and $$ \dfrac{C}{Dx^{\alpha+\beta}}$$ The form i want $$ \dfrac{E}{Fx^{\alpha+\beta+1}}$$ I was thinking to do partial fractions or ...
3
votes
1answer
25 views

Partial Fraction Decomposition with exponent in numerator

$$Y(s)=\frac{1-e^\frac{-\pi*s}{2}}{s^2[(s+\frac{1}{2})^2+1]}$$ This is actually a step in an differential equations problem. I need to decompose this so I can solve the ODE. I know how to solve ...
2
votes
2answers
50 views

Proof by induction, or without it if possible?

I was given a task to prove: $$ \frac{1}{(x+1)(x+2)\ldots(x+n)}=\frac{1}{(n-1)!}\sum_{i=1}^n\binom{n-1}{i-1}\frac{(-1)^{i-1}}{x+i} $$ I am almost 100% sure this is best solved by induction but to be ...
1
vote
1answer
30 views

partial fraction problem

I have thought for a long time but general method is not working : $$\frac{s^2+2}{(s+2)^2(s^2+2s+2)^2}$$ Thanks for all help.
0
votes
0answers
25 views

Partial fractions in Laplace Transform

Solve: $$y''+y'+\frac{5}{4}=U_\frac{\pi}{2}(t)f(t-\frac{\pi}{2})$$ becomes: $$[s^2+s+\frac{5}{4}]Y(s)=\frac{1-e^\frac{-\pi*s}{2}}{s^2}$$ becomes: ...
1
vote
2answers
46 views

Evaluating $\int [(5x^3-3x^2+7x-3)/(x^2+1)^2] dx$

Can I get hints on how to factor? Should I break down by term, i.e. $$\frac{5x^3}{(x^2+1)^2}\text{...?}$$
0
votes
1answer
22 views

Factor to solve partial fractions decompositions

$\int$$ (x^2-x-21)dx\over(2x^3-x^2+8x-4))$ I know how to factor the denominator but I don't know how to factor the numerator. Could I get a step by step breakdown of how to solve??
0
votes
3answers
31 views

Partial fraction decomposition of rational functions

How can I start this with Partial fractions? $$ \frac{10}{(s^2-4)(s^2+4)}+\frac{1}{s^2+4}$$ I was thinking of something like: ...
0
votes
2answers
47 views

Is this a quadratic factor or a repeated linear factor?

If I need to integrate something like $\frac{1}{y^2(1-y)}$, and I use the method of partial fractions, how do I know whether the $y^2$ in the denominator is a quadratic factor or just a repeated ...
17
votes
5answers
1k views

Can all real polynomials be factored into quadratic and linear factors?

So I understand how to do integration on rational functions with a linear and a quadratic denominator, and I understand how to do a partial fraction decomposition, but I was wondering what happens if ...
0
votes
3answers
57 views

Factoring a polynomial of fourth degree with false roots: $x^4+4$

I want to write this polynomial in factored form: $$x^4+4$$ The reason I want to do this is to be able to make partial-fraction decomposition on it to make an integrand easier to integrate. What's ...
1
vote
3answers
63 views

easier way to decompose fraction into partial fraction

It is a question in a test, and I couldn't manage to complete it. Given a complex fraction $\frac{1}{(z-1)^3(z+1)^3}$, we know that it can be decompose into ...
0
votes
0answers
13 views

Partial Fraction Decomposition — Inverse Laplace Transforms

I apologize if this is a rather lame question, but I've always been a little touchy with my partial fraction decompositions and I'm hoping to get better at them. Could you verify (or correct?) my ...
0
votes
2answers
19 views

What formula do I use when i have to find the partial fraction of

What formula do I use when i have to find the partial fraction of $$\frac{10x^2+11x+19 }{ (x-0.5)(2x^2+6x+10)}$$ Is it $A(2x^2+6x+10) + (Bx+C)(x-0.5)$ ? Or do I have to factorise $(2x^2+6x+10)$ ...
-1
votes
1answer
22 views

Evaluate the integral (using partial fractions maybe?) [duplicate]

Evaluate the following integral $\int{\frac{1}{(x+a)(x+b)}}$ (this might involve partial fraction decomposition, $\int{\frac{1}{x^2+x(a+b)+ab}}$ this is what my first step was)
1
vote
2answers
28 views

Partial fraction of $\frac{2x^2-9x-9}{x^3-9x}$

I'm doing some questions from Anton, 8th edition, page 543, question 13. I've found a answer but it does not match with the answer given at the last pages. Questions asks to solve ...
1
vote
1answer
18 views

Correct partial fraction construction?

Is the below the correct partial fraction decomposition? $$\frac{s^2 - 6s + 9}{(s-2)^3}=\frac{A}{s-2}+\frac{B}{(s-2)^2}+\frac{C}{(s-2)^3}$$ I can see that the numerator doesn't have a factor of ...
1
vote
2answers
48 views

Shortcut methods for Partial fraction decomposition in IVPs solved by Laplace transformation?

I have an IVP I'm trying to solve with Laplace transformations: $$y''-4y'+4y=te^{2t}$$ Given that: $y(0)=1$ and $y'(0)=0$ I've gotten to the part where I isolate $Y(s)$: ...
0
votes
1answer
23 views

Is this the correct setup for partial fractions? $\frac{1-e^{-s} + se^{-s} + s^3}{s^2(s^2+2)}=\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+2}$

I am trying to inverse laplace transform the following: $$F(s)=\frac{1-e^{-s} + se^{-s} + s^3}{s^2(s^2+2)}$$ and I believe what I do is take: $$\frac{1-e^{-s} + se^{-s} + ...
0
votes
1answer
20 views

Partial Fractions to solve Logistic Equation

I am not really understanding how my book is getting $$\frac{x'}{x(1-\frac{x}{K})}=\frac{x'}{x}+\frac{x'}{K-x}$$ so $$\frac{x'}{x(1-\frac{x}{K})}=\frac{x'}{x-\frac{x^2}{K}}$$ ...
1
vote
2answers
63 views

Writing $\frac{x^4(1-x)^4}{1+x^2}$ in terms of partial fractions

How does one write $$\frac{x^4(1-x)^4}{1+x^2}$$ in terms of partial fractions? My Attempt $$\frac{x^4(1-x)^4}{1+x^2}=\frac{x^4-4x^5+6x^6-4x^7+x^8}{1+x^2}$$ ...
1
vote
1answer
26 views

Laplace transform convolution

$x(t) = cos(3πt)$ h(t) = $\exp(-2t)u(t)$ Find y(t) = x(t) * h(t) (ie convolution) Y(s) = X(s)H(s) and then take inverse laplace tranform of Y(s) $ L(x(t)) = \frac{s}{s^2+9π^2} $ $ L(h(t)) = ...
1
vote
1answer
22 views

Integration $\int\frac{\sqrt{x+4}}x dx$ by partial fraction

Here's what I came up with: $$\int\frac{\sqrt{x+4}}x dx$$ for $ u = \sqrt { x + 4 } $ $$=\int\frac{u}{u^2-4}\;2u\;dx$$ $$=2\int\frac{u^2}{(u-2)(u+2)}\;dx$$ $$=\frac{A}{u-2}\;+\;\frac{B}{u+2}$$ ...
1
vote
3answers
28 views

How to convert this polynomial to partial fraction?

I want to convert this polynomoial to partial fraction. $$ \frac{x^2-2x+2}{x(x-1)} $$ I proceed like this: $$ \frac{x^2-2x+2}{x(x-1)} = \frac{A}{x} + \frac{B}{x-1} $$ Solving, $$ A=-2,B=1 $$ But ...
1
vote
1answer
21 views

Find $\sum^{10}_{r=1}\frac{1}{(3r-1)(3r+2)}$ and its $\sum^{\infty}_{r=1}$

Since $$ \sum^{n}_{r=1}\frac{3}{(3r-1)(3r+2)}=\frac{1}{2}-\frac{1}{3n+2} $$ find the sum of the series $$ \frac{1}{2\times5}+\frac{1}{5\times8}+\frac{1}{8\times11}+\cdots+\frac{1}{29\times32} ...
0
votes
0answers
37 views

Notes on theory of partial fraction decomposition

I tried searching a lot but mostly I am seeing techniques on how to decompose polynomial denominators. What I am looking for is the theory that helps me get a total picture. For example, on this link ...
0
votes
2answers
22 views

Partial Fractions (3 Factors)

This document outlines a shortcut for partial fractions involving 2 factors in the denominator (P(x) + a) and (P(x) + b). At the end of the document it gives a challenge to find a similar shortcut ...