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

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1answer
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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 ...
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3answers
29 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 ...
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1answer
44 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 ...
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0answers
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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. ...
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1answer
17 views

Partial fraction decomposition coefficients problem [closed]

How can I go about integrating $\int\frac{dx}{x(x-1)(x^2+4)^2}$ What are my factors?
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1answer
26 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) = ...
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0answers
26 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: $\frac{3 z^6-10 z^5+6 z^4+10 z^3-6 z^2+3z}{z^7-3 z^6+2 z^5+36 z^2-51 z+15}$ ...
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1answer
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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.
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21 views

Easy partial fraction decomposition

I need to decompose the rational function $R_z$ as a partial fraction as shown on the second line. How do I begin?
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1answer
37 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 $C$ ...
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2answers
40 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 ...
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1answer
22 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 ...
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2answers
49 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 ...
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1answer
28 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.
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0answers
23 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: ...
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2answers
45 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. $$\frac{5x^3}{(x^2+1)^2}\text{...?}$$
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1answer
20 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??
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3answers
30 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: ...
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2answers
40 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 ...
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4answers
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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 ...
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3answers
55 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 ...
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3answers
59 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 ...
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0answers
12 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 ...
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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 $(2x^2+6x+10)$ ...
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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)
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2answers
27 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 ...
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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 ...
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2answers
32 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 $Y(s)$: ...
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1answer
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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} + ...
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1answer
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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 $$\frac{x'}{x(1-\frac{x}{K})}=\frac{x'}{x-\frac{x^2}{K}}$$ ...
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2answers
60 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 $$\frac{x^4(1-x)^4}{1+x^2}=\frac{x^4-4x^5+6x^6-4x^7+x^8}{1+x^2}$$ ...
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1answer
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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)) = ...
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1answer
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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 } $ $$=\int\frac{u}{u^2-4}\;2u\;dx$$ $$=2\int\frac{u^2}{(u-2)(u+2)}\;dx$$ $$=\frac{A}{u-2}\;+\;\frac{B}{u+2}$$ ...
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3answers
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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 ...
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1answer
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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} ...
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0answers
36 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 ...
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2answers
21 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 ...
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2answers
590 views

Integral of rational function with higher degree in numerator

How do I integrate this fraction: $$\int\frac{x^3+2x^2+x-7}{x^2+x-2} dx$$ I did try the partial fraction decomposition: $$\frac{x^3+2x^2+x-7}{x^2+x-2} = \frac{x^3+2x^2+x-7}{(x-1)(x+2)}$$ And: ...
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2answers
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Expressing $\frac{1}{4n^2-1}$ as a partial fraction

I was asked to express $$\frac{1}{4n^2-1}$$ as a partial fraction. I have no clue as to what I should break this into. For example I know : $$\frac{1}{n(n-1)}= \frac {A}{n} + \frac {B}{n-1}$$ ...
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3answers
34 views

How do I partial fraction this

I have this fraction that I want to express as partial fractions: $$\frac{s}{(s^2+1)(s-1)}$$ How do I do it? I came as far as the expression: $$s=A(s-1)+B(s^2+1)$$ But how do I solve this for A ...
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3answers
243 views

Partial fraction decomposition of a rational function with denominator $x^5+2x^4+x^3-x^2-2x-1$

Factorize the denominator completely and write $f(x)$ as a partial fraction given $$f(x) = \frac{2x^5+15x^4+15x^3+2x^2+2}{x^5+2x^4+x^3-x^2-2x-1}$$ Any ideas for this partial fraction question? ...
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2answers
35 views

The integral of $x^3/(x^2+4x+3)$

I'm stumped in solving this problem. Every time I integrate by first dividing the $x^3$ by $x^2+4x+3$ and then integrating $x- \frac{4x^2-3}{x+3)(x+1)}$ using partial fractions, I keep getting the ...
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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 ...
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1answer
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Finding partial fractions [closed]

I'm attempting to split the fraction $$\frac{3x^2+6x+2}{(2x+3)(x+1)^2}$$ into partial fractions. Any help would be greatly appreciated.
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1answer
46 views

integrating method (maybe PFD)

I am trying to integrate: dt = 1/(ax-bx^2) * dx I am guessing I need to use Partial Fraction decomposition, can someone help show me how to begin this process?
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Is this the correct procedure for Integral Partial Fraction.

$∫ (x^3+x^2+x+3)/((x^2+1)(x^2+3))$ The first step I did is distributed the denominator so that I can find out if I should use synthetic division. Which after doing this I discovered that the ...
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2answers
20 views

Partial Fraction Decomposition Problem

I am having trouble with this problem. I need to integrate: $$\frac1{T^4}\times \frac1{K-T}$$ with respect to $T$. If I do PFD: $$\frac{A}{T^4} + \frac{B}{T^3} + \frac{C}{T^2} +\frac{D}{T} + ...
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4answers
35 views

Taking partial fractions for integration?

I'm having some trouble with integrals involving partial fractions it seems. Been stuck on this forever. The equation is given below and I have to use partial fractions to solve. ...
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0answers
45 views

Partial fraction help

I need Help figuring out how to solve the indefinite integral of $$\int{ -5x^3-2x^2+32\over x^4-4x^3 } dx $$ using partial fractions. Please help. Thank you! I have already checked the online ...
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3answers
31 views

Starting a Partial Fractions Question

I have the question $$ \frac{ 3x + 3 }{ (x-1)(x^2 +x +1) } $$ and I am unsure about how to start as the quadratic on the denominator is irreducible. So anyone got any tips for starting this one?