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

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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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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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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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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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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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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? :)
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Partial fractions for $\pi \cot(\pi z)$

I want to derive $$\pi \cot(\pi z) = \sum_{-\infty}^{\infty}\frac{1}{z-n} + \frac{1}{n}$$ without taking derivatives. I know through Mittag Leffler that $$\pi \cot(\pi z) = g(z) ...
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Partial Fractions with an irreducible quadratic factor

$\int \frac{2}{(x-4)(x^2+2x+6)} dx$. this is a partial fraction with irreducible quadratic factors. I know how to set it up and I found A, B, and C. 2 = A((x^2)+2x+6) +(x-4)(Bx+C). then I plugged ...
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Using Laplace transforms to solve a convolution of two functions

Hi I have this problem where I need to take the convolution of functions and I am not sure if I got the right answer or something close so any advice or help would be very appreciated. So here is the ...
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Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: ...
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How to Decompose into Partial Fractions

Why is it useful to write: $$\frac{}{(x+1)^2(x-1)}=\frac{}{(x+1)^2}+\frac{}{x+1}+\frac{}{x-1}$$ and not: $$\frac{}{(x+1)^2(x-1)}=\frac{}{(x+1)^2}+\frac{}{x-1}$$ when decomposing into partial ...
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complex conjugate pairs of a quartic

I tried my hand at this question, which included finding the partial fractions of $\frac{x^2}{1-x^5}$. I found a factor of $1-x$ for the denominator, but I do not know how to work out the complex ...
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How to evaluate $\int \frac{x^6}{x^4-1} \, \mathrm{d}x.$

Evaluate the integral: $$\int \frac{x^6}{x^4-1} \, \mathrm{d}x$$ After a lot of help I have reached this point: $x^2 = Ax^3 - Ax + Bx^2 - B + Cx^3 + Cx^2 + Cx + C + Dx^3 - Dx^2 + Dx - D$ But now I ...
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Partial fraction that contain special function

How to apply partial fraction to the following equation: $$ \frac{e^{\frac{(2c+5x)}{3x}} \mathop{E_{n}}\nolimits\!\left(x\right)}{(a+x)(b+x)} $$
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How to simplify this mathematical expression?

I found this: Let's rewrite the integrand so that it's easier to integrate: $$\dfrac{x}{(x-2)^2(x+1)} = -\dfrac{1}{9x+9}+\dfrac{1}{9x-18}+\dfrac{2}{3(x-2)^2}$$ This is the mathematical ...
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Integrating via partial fractions

My question is this: Why is it that fractions have to be split up in a very specific manner? For example if I have $\frac{5x}{(x+1)^2}$ this fraction HAS to be split up like this:$$\frac ...
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Integral of $\frac{x^2}{x^2-4}$

How would I integrate the following: $$\int \frac{x^2}{x^2-4}\ dx$$ We have covered three techniques for integration: substitution, integration by parts and partial fractions. I have tried partial ...
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How does Wolfram get from the first form to the second alternate form?

So, I was trying to compute an integral but I couldn't actually manage getting anywhere with it in its initial form. So, I inserted the function in Wolfram Alpha and I really got a nicer form (second ...
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Partial fraction (doubt)

I have this partial fraction $$\displaystyle\frac{1}{(2+x)^2(4+x)^2}$$ I tried to resolve using this method: ...
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53 views

Faster method for partial fractions

Is there a way to apply the "cover-up" method when solving for fractions of the following type? $$\frac {2x}{(x+1)(x^2+1)^2}$$ The long way would be; $$\frac {2x}{(x+1)(x^2+1)^2} = \frac {A}{x+1} + ...
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Help w/ Partial Fraction Decomposition

I need some help figuring out how to decompose $\displaystyle\frac{1}{x^4+1}$ into partial fractions. This is what I have done so far: $$\frac{1}{x^4+1} = \frac{1}{(x^2 - \sqrt{2}x + 1)(x^2 + ...