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

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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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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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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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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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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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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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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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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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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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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
23 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)) = ...
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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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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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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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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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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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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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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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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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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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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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28 views

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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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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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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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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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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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?
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Different methods of Partial Fractions.

By the method of partial fractions we take: $$\frac{px+q}{\left(x-a\right)\left(x-b\right)}=\frac{A}{x-a}+\frac{B}{x-b}$$ ...
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Differential equation $x'=11x -x^2 -24$

Getting stuck finding $x(t)$ on the differential equation: $dx/dt = 11x -x^2 -24$ with $x(0)=5$. So my work so far is: $dx/(11x-x^2-24) = dt$ Using partial fractions $A(x-3) + B(x-8) = 1$, so $A = ...
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Partial fraction expansion of $1/(e^x-1)$

I'm reading a paper by Riesel and Gohl; in it they say that "partial fraction expansion" of $1/(e^x-1)$ is $$\frac{1}{e^x-1}=\frac{1}{x}-\frac{1}{2}+2x \sum_{k=1}^{\infty} \frac{1}{x^2+4\pi^2k^2}$$ I ...
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Partial Fraction Expansion in MATLAB

I am to use MATLAB to find the partial fraction expansion of the following function. Can this be done in that format or do I have to manipulate the function? Note: This must be done using pure MATLAB ...
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Rewriting $1/(v^2-(mg/k))$ as two fractions

I'm looking at the solution someone gave to me and I'm having a bit of trouble following one of the steps. This step in particular, ...
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Partial fractions and Hermit reduction?

According to the late Manuel Bronstein in his "Symbolic Integration Tutorial" (ISSAC 98), and available here, the partial fraction algorithm "should not be used in practice, yet it remains the method ...
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Partial Fraction Decomposition, trivial mistake

I'm doing a trivial mistake for sure, but I struggling to find it... I have: $$ \frac{160}{4s^2+4.8s+4} = \frac{160}{(s+0.6-0.8i)(s+0.6+0.8i)} = \frac{K_1}{s+0.6-0.8i} +\frac{K_1^*}{s+0.6+0.8i} $$$$ ...
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Struggling with $\int \frac{dy}{y\left(1 - \frac y2\right)}$

I know I need to use a partial fraction and suspect I will end up with 2 terms that end up as a natural log integral but I just can't work it out. $$\int \frac{dy}{y\left(1 - \frac y2\right)}$$ I ...
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Basic partial fractions issue

I've been starting to get the hang on partial fractions, whilst I've been able to do most of the basic ones, this kept causing some issues so I assumed: I'm using the wrong method I'm converting ...
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Difference in writing answer for $\int\dfrac{12x^3-24x^2+5}{x^2-2x}dx$

I have this integral: $\int\dfrac{12x^3-24x^2+5}{x^2-2x}dx$ I solved using partial fractions and got an answer: $6x^2-\frac{5}{2}\ln(x)+\frac{5}{2}\ln(x-2) +C$ But I am using "My Math Lab" online ...
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1answer
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Laplace Transform assistance

Find the inverse laplace transform of: $\frac{25}{(s-1)^2(s^2+4)}$ $\frac{25}{(s-1)^2(s^2+4)}=\frac{A}{s-1}+\frac{B}{(s-1)^2}+\frac{C}{s^2 + 4}$ $$25=A(s^2+4)(s-1)+B(s^2+4)+C(s-1)^2$$ ...
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Partial fraction in two variables

I want to find a partial fraction expansion for the following: ($b$ is a constant) $$\frac{1}{(b^2+x^2)(b^2+y^2)}$$ As there are two variables, I am unsure what form the decomposition should be of. ...
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help in Laplace and partial fractions

Can any one teach me how to solve C2.(a) and (b) step by step? C2. (a) Resolve $\frac{1}{s^2(s^2+s+1)}$ into partial fractions of the form $\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+s+1}$. Hence, ...
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Using partial fraction for inverse Laplace transform of $1/[s(s+5)^2]$

my question is the last part $1/5(s+5)^2$, how is it become $-5te^{-5t}$ I thought is should be -$1/5 te^{-5t}$
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Shorter way to integrate $\int \frac{x^9}{(x^2+4)^6} \, \mathrm{d}x$

$$ I=\int \frac{x^9}{(x^2+4)^6}\mathrm{d}x $$ Yeah I know, I can substitute: $$t=x^2+4\text{ or }2\tan\theta$$ So that: $$I=\frac12\int\frac{(t-4)^4}{t^6}\mathrm{d}t\text{ or } ...
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Partial fraction decomposition of type $1/(x^2+k)$

I know that partial fraction of this can be written as: $$\frac{3x}{(1+x)(2+x)}=\frac{-3}{1+x}+\frac{6}{2+x}$$ Which can be done in these ways: ...
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79 views

How to integrate $\frac{x+4}{x^2+2x+5}$

Having a little trouble on how to break it up. How to integrate $$\frac{x+4}{x^2+2x+5}$$
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1answer
46 views

Inverse Laplace transform (using table) when denominator cannot be factored

Usually when performing inverse Laplace transforms, I decompose the function into partial fractions, and then look up standard transforms in a table. For example: $$Y(s) = ...
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1answer
39 views

Partial fractions where the denominator is one function

I need to solve this differential equation for x: $$ \frac{dv}{dx} = \frac{4000}{v} - 0.9v $$ Rearranging: $$ \frac{dx}{dv} = \frac{1}{4000v^{-1} - 0.9v} $$ How would I go about splitting this ...
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Decompose a fraction in a sum of two

Let's say that I have this fraction: $$ \frac{2x}{x^2+4x+3}$$ I would like to decompose in two fraction: $$ \frac{A}{x+3} + \frac{B}{x+1}$$ Which is the procedure for that? :)