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Questions tagged [partial-fractions]

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

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How do I decompose this faction into partial fractions?

Here is the fraction: $$\frac{1}{(ar+1)(ar+a+1)}$$ I looked at the mark scheme, and it says the answer is: $$\frac{1}{a}(\frac{1}{ar+1}-\frac{1}{ar+a+1})$$ but when I tried it I got: $$\frac{1}{ar+1}-\...
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Calculate a difficult integral resulting from a partial fraction decomposition [closed]

How do you calculate such an integral ? $$ \int \frac{Ex +F }{ (x^2 + 1)^2 } dx $$
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Decomposing a Fraction Involving Cube Roots for Integration

I had an exam the other day and there was this question to decide whether the following function is improperly integrable from 0 to 1. I wrote a solution for it but now I came to understand it's not ...
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Partial Fractions of $\frac{1}{(x+1)^{m}(x+3)^{n}}$

In the partial fractions decompostion of $$\frac{1}{(x+1)^{m}(x+3)^{n}}$$ Is there any general formula for the coefficient of $\frac{1}{x+3}$ term? Of course, I am able to get the coefficients ...
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Avoiding even powers in partial fraction decomposition

Consider the expression: $$\frac{1}{(x-a_1)^{n_1} (x-a_2)^{{n_2}}}$$ where $a_1$ and $a_2$ are real numbers and $n_1 \geq 2$ and $n_2 \geq 1$ Am I correct that it is impossible to decompose this in a ...
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Write $\frac{\sqrt{x+a}}{x+b}$ as a series of fractions of the form $\frac{1}{x+c}$ or $\frac{1}{x(x+d)}$?

I need to approximate the function $\frac{\sqrt{x+a}}{x+b}$ as a series of fractions of the type $\frac{1}{x+c}$ or $\frac{1}{x(x+d)}$, i.e. $$\frac{\sqrt{x+a}}{x+b}=\sum_{n=1}^\infty \left(e_n\frac{1}...
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Partial fractions with a repeated factor [duplicate]

I am looking to find a derivation, or a intuitive explanation as for why a partial fraction with a repeated factor needs to include a factor in the expansion for each power possible. How does one ...
James Chadwick's user avatar
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Determining denominator of partial fractions

Before, integrating, we can often split a fraction into its partial fractions to make the integration process significantly more simple. However, I have realised that this fraction we can split can ...
James Chadwick's user avatar
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1 answer
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Justification for equality in partial fraction expansion from generatingfunctionology by Herbert S. Wilf

The problem is from generatingfunctionology by Herbert Wilf on page 4. My question is not about the process of getting the generating function (they do a good job in this post) but rather where the ...
David Farmilant's user avatar
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How to compute the partial fraction decomposition of $\frac{6}{x^4(x+1)}$?

How do I compute the partial fraction decomposition of $\frac{6}{x^4(x+1)}$? I let $$\frac{6}{x^4(x+1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4} + \frac{F}{x+1}$$ When I let $x=-1$...
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how to divide $a^{n-1} x^n+x((-1)^n+a^n)+a^{n-1}$ by $ax+1$?

I need to find $a(n,k)$ where $$ \sum_{k=0}^{n-1}a(n,k) x^k=\frac{1}{a^n+(-1)^n}\frac{a^{n-1} x^n+x((-1)^n+a^n)+a^{n-1}}{ax+1}$$ where I need it to find a general partial fraction form $$ \frac{x}{(x^...
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Partial fraction decomposition for $\frac{1}{(z^2+\pi^2)^2}$?

How can I compute partial fraction decomp for $\frac{1}{(z^2+\pi^2)^2}$? I, for some reason, am not able to do it. Here's what I tried: $$\frac{1}{(z^2+\pi^2)^2} = \frac{1}{(z+i\pi)^2(z-i\pi)^2} = \...
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How can i do the following partial decomposition?

I need to prove that: $$ \frac{1}{(x-a)(x-b)} = \frac{1}{(b-a)(x-b)}- \frac{1}{(b-a)(x-a)}, $$ and I must note that I need to go from the left expression to the right (because of the exercise). So, I ...
Miguel Simões's user avatar
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Finding the Inverse Laplace transform using partial fractions

Problem: Given: $$ Y(s) = \dfrac{3s^2 + 6s+ 84} {( s+1 )(s-2)(s^2+ 2s+10) } $$ Find $y(t)$ by computing the inverse Laplace transform. Answer: To do this, we use the technique of partial fractions. \...
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Closed form of $a_{n+2}=a_{n+1}a_n+1$

I was given this sequence and I need to find a closed form. $$a_0=1,a_1=2$$ $$\text{and } \forall n \geq0\text{ } a_{n+2}=a_{n+1}a_n+1$$ I tried defining the following generating function: $$A(q)=\...
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General partial fraction decomposition for a specific type of rational function [closed]

Given a rational function of the form $$ \frac{x^k}{(x^n-\lambda_1)(x^m-\lambda_2)}$$ with $k< n+m$, I know we can prove that there are unique polynomials $p(x),q(x)$ with $$ \frac{x^k}{(x^n-\...
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integration of$ f(z)=\frac{1}{z^4+1}$ using Cauchy's Integral Formula vs. Partial Fractions

I get different answers when I try to evaluate the following integral when I use partial fraction decomposition and Cauchy's Integral Formula. The integral is, $$\color{blue}{\int _{-2}^{2}\frac{1}{z^...
Travis Miller's user avatar
2 votes
2 answers
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How to show that two generating functions share congruent coefficients

I have two generating functions. $A(x)= \frac{x^{\frac{m \left(m-1\right)}{2}}}{\left(1-x\right)\cdot \left(1-x^{2}\right)\cdot ...\cdot \left(1-x^{m}\right)}$ $B(x)= \frac{A(x)}{x^t} = \frac{x^{\frac{...
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Textbook says to integrate a fraction using 'Taylor's formula'?

I don't understand the solution my textbook gives for this problem: $$ \int \! \frac{x^3}{(x+1)^5} \, \mathrm{d}x $$ I thought it had to be done with partial fractions, but I couldn't get it right, ...
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Can someone help me with partial fraction expansion, please? [closed]

$$\frac{1}{(1-x^a)(1-x^b)}=\frac{A}{(1-x)^2}+\frac{B}{(1-x)}+\sum_{r^a=1}^{ ‎ }\frac{C_r}{(1-x/r)}+\sum_{t^b=1}^{ ‎ }\frac{D_t}{(1-x/t)}$$ $(t,r\neq1; A, B, C, D$ are real numbers) How did the author ...
ChemistryLearner's user avatar
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I found an interesting question but I keep getting stuck in a loop. [closed]

Find all values of A, B, C and C such that: $$ \frac{x-1}{(x-1)(x-2)(x-2)} = \frac{A}{x+1} + \frac{B}{x+2} + \frac{C}{(x-2)^2} $$ I keep getting into a loop in which: $$ x - 1 = Ax^2 - 4Ax + 4A + Bx^2 ...
Durian's user avatar
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Partial Fractions in Integrations with complex Factorization.

I know how to integrate $\sqrt{tan}$, using the "classical approach" which I know has been illustrated on another question before. However, I have just recently learnt to factorise using ...
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Converting a proper fraction into partial fraction

For solving integration-related questions, a rational proper fraction of the form $\frac{px^{2}+qx+r}{(x-a)(x^{2}+bx+c)}$ is decomposed into the sum of the expressions, $$\frac{A}{x-a} + \frac{Bx+C}{x^...
Sasikuttan's user avatar
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If the third derivative of $\frac{x^4}{(x-1)(x-2)}$ is $\frac{-12k}{(x-2)^4}$ + $\frac{6}{(x-1)^4}$ then the value of k is? [closed]

In the answers I found on google, see this link, they converted the given function into a certain form? What is the process of that conversion (I understand it is a partial fraction of sorts, but how ...
Sanchita's user avatar
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Partial fraction decomposition of derivative over polynomial

We know about standard partial fraction decomposition that says if $f$ and $g$ are two non-zero polynomials over a field $K$ with $g = \displaystyle\prod_{i=1}^k p_i^{n_i}$ being a product of ...
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Laurent Series around $z_0=0$ and $z_0=1$

I tried to expand three functions in Laurent series, but I don't have any given answer for them and I'm not very confident that what I'm doing is correct. The first one is $f(z)=\frac{z^3e^{1/z}}{z+1}$...
poxipollepi1's user avatar
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Partial fraction decomposition of $\frac {2x^2 + x}{(x+1)(x^2+1)}$

Problem: Expand $$\frac {2x^2 + x}{(x+1)(x^2+1)}$$ using partial fraction decomposition. I wrote the following $$\frac A {x+1} + \frac B {x^2 + 1} = \frac {x(2x+1)}{(x+1)(x^2+1)} \\ A(x^2+1) + B(x+1) =...
SRobertJames's user avatar
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Show that $1+\sum_{n=1}^{\infty} \frac{1}{n^2 (n+1)} = \frac{\pi^2}{6}$

Show that $$1+\sum_{n=1}^{\infty} \frac{1}{n^2 (n+1)} = \frac{\pi^2}{6}$$ Proof By Partial Fraction Decomposition $$\frac{1}{n^{2} \left(n + 1\right)}=\frac{-1}{n}+\frac{1}{n^{2}}+\frac{1}{n + 1}$$ ...
vengy's user avatar
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2 answers
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Integration help for a beginner

This might be a silly question but I am a beginner in calculus so I really do not understand. When solving the problem $\int\frac{5+x}{\sqrt{16−(x+4)^2}}dx$, why we can’t divide it into the two ...
Seifeldeen Abdullatif's user avatar
5 votes
4 answers
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Extracting the sequence generated by $\frac{1}{(1-x)(1-x^2)(1-x^3)}$ [duplicate]

So I need the general formula for the sequence generated by the generating function $$ \frac{1}{(1-x)(1-x^2)(1-x^3)}. $$ My idea was the decompose this into partial fractions and thus easily deduce ...
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6 votes
3 answers
296 views

Computing $ I_n=\int_0^1 \frac{x^n}{6+x-x^2} d x $

After reading the reduction formula in the post, I am curious about the closed form of the integral $$ I_n=\int_0^1 \frac{x^n}{6+x-x^2} d x $$ I first resolve the integrand into two partial fractions ...
Lai's user avatar
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3 votes
5 answers
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I need guidance in this integral

$$\int \frac{e^{2x}-2}{e^{2x}+7}\,\Bbb dx$$ I was trying: $$\int \frac{e^{2x}}{e^{2x}+7}\,\Bbb dx -2\int \frac{1}{e^{2x}+7}\,\Bbb dx$$ $$= \frac{1}{2}\ln|e^{2x}+7| -2\int \frac{1}{(e^x)^2+(\sqrt{7})^2}...
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I get stuck with this integral

$$\frac{1}{10} \int \frac{3x+20}{x^2+10}dx$$ I was trying to factor out the denominator: $$\frac{1}{10} \int \frac{3x+20}{x^2+(\sqrt{10})^2}dx$$ $$\frac{1}{10} \int \frac{3x+20}{(x+\sqrt{10})^2-2x\...
samsamradas's user avatar
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Integrate reciprocal of polynomial (coefficient of partial fraction decomposition)

I am working on the integral of the reciprocal of polynomial such that $\int{\frac{x^{q-1}dx}{x^{p}-ax^{q}+1}}$ where $p$ and $q$ are coprime integers. I tried to solve it by partial fraction ...
Summer's user avatar
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Complex exponential, getting constants from partial fraction decomposition in Laplace domain (used s = j*omega) prove / explain please

I came across this in a control engineering textbook: consider the transfer function $G(s)$ as the following ratio of functions, where the denominator is a polynomial in s: $$G(s) = \frac{p(s)}{q(s)} =...
Mr Phase Locked Loop's user avatar
2 votes
1 answer
116 views

Why Partial Fractions Decomposition works [duplicate]

Consider this partial fraction. $$\dfrac{x-25}{x^2+5x-24}=\dfrac{A}{x-3}+\dfrac{B}{x+8}$$ Multiply both sides by the quadratic $x^2+5x-24$. $$x-25=A(x+8)+B(x-3)$$ From here, I've seen many textbooks ...
Anirudh Yamunan Govindarajan's user avatar
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Partial fraction decomposition over $\mathbb{F}_{n}[x]$

Consider the rational function $\dfrac{f(x)}{g(x)} \in \mathbb{F}_{n}[x]$ such that $g(x) = p(x) \cdot h(x)$. If $\gcd\left(p(x), h(x)\right) = 1$, then $$\dfrac{f(x)}{g(x)} = \dfrac{f(x)}{p(x) \cdot ...
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How do I solve a partial fractions question when $x$ can’t equal any solutions?

How do I solve $A(i)$ and $A(ii)$ when $x$ can't equal $-2/5$ or $1/2$? The only method I’m aware of for partial fractions requires these values to be subbed in and I’m confused. Sorry if the image is ...
Lorcan Kearney's user avatar
3 votes
3 answers
375 views

how to evaluate $ \int \frac{3x^5 -x^4 +2x^3 -12x^2-2x+1}{(x^3-1)^2} dx$?

I saw the following problem $$ \int \frac{3x^5 -x^4 +2x^3 -12x^2-2x+1}{(x^3-1)^2} dx$$ I tried to solve it by partial fractions and after 7 pages of a lot of calculations I was able to prove that $ \...
pie's user avatar
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Can someone run me through using heaviside cover-up for this decomposition

Can someone run me through using Heaviside cover-up for this decomposition: $$ \frac{2}{(x-2)(x+3)(x+1)^3} $$ I have calculated the decomposition traditionally but can't get consistent answers using ...
trawling's user avatar
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Difficulty With An Inverse Laplace Transform

I am a physics student modeling a quantum mechanical system for my undergraduate research. I am currently solving for the time-dependent coefficients $c_1(t), c_2(t), c_3(t)$ on a super position of ...
AD203's user avatar
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1 vote
3 answers
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Integrating a real-valued function with complex numbers

I want to integrate $$\int\frac{x}{x^2+4x+5}\;dx$$ and I tried to use partial fractions because $x^2+4x+5=(x+2+i)(x+2-i)$. Therefore: \begin{align} \frac{x}{x^2+4x+5}&=\frac{A}{x+2+i}+\frac{B}{x+2-...
Fynn Zentner's user avatar
3 votes
2 answers
141 views

Partial fraction of integral giving no solution.

The below integral is to be integrated using partial fraction decomposition $$\int \frac{3x + 5}{(x-1)^2(x+1)} \, dx$$ I tried to form a partial fraction in the following manner and ended up getting a ...
Nitish's user avatar
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Applying partial fraction decomposition when evaluating a contour integral

Take for example the integral in (1) where $\gamma$ is the anticlockwise path around the top half of the circle $|z|=r$ According to Wolfram, $$\int \frac{1}{z+w}dz=\operatorname{Log}\left(z+w\right)+...
Simon M's user avatar
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How to inverse z complex transform?

I have this z transform $$32(\frac{z(z+\frac{3}{8})}{(z-z_{1})(z-\overline{z}_{1})})$$ with $$z_{1}=\frac{1}{16} + \frac{1}{4}i$$ The problem is I am not sure how to perform a partial fraction on ...
ggreg's user avatar
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2 votes
1 answer
371 views

Partial fractions trick, repeated roots [closed]

Do you know how can one extend this trick to find partial fractions coefficients when the roots of the denominator are repeated? From now, I'm just interested in the cases when the roots are algebraic....
Daniel Checa's user avatar
2 votes
2 answers
256 views

Partial fraction decomposition of $\frac{1}{(x(x+1)(x+2)...(x+n))^2}$

In view of this question, I am trying to find the partial fraction decomposition of $$\frac{1}{(x(x+1)(x+2)...(x+n))^2}$$ where $n\in\mathbb{N}$ Since every $k$, $k=-n,...,-2,-1,0$ is a pole of order ...
Max's user avatar
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1 vote
2 answers
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Contour integral over function $P(x)/Q(x)$: $P(x) = 1$ and $Q(x)$ can be broken into linear factors

a. Let $z_1,z_2,...,z_n$ be distinct complex numbers $(n \geq 2)$. Show that in the partial fractions decomposition \begin{equation} \frac{1}{(z-z_1)(z-z_2)\cdots(z-z_n)} = \frac{A_1}{z-z_1}+\frac{A_2}...
Alex's user avatar
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Can one always use complex numbers for finding the coefficients of a partial fraction decomposition?

I have following term, which I would like to decompose into partial fractions: $ \frac{1}{x^2(1+x^2)} \overset{!}{=} \frac{Ax+B}{1+x^2} + \frac{C}{x} + \frac{D}{x^2}$ Multiplying everything out gives:...
haifisch123's user avatar
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1 answer
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how does the method of partial fraction decomposition by long division works when there is repeated root?

I am reading A treatise on integral calculus and I can't understand what does the author means here, how does $\frac{y^2+2y+1}{y+2}$ turned into $0.5 +0.75y+0.125y^2-\frac{y^3}{8(2+y)}$ and I didn't ...
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