Questions tagged [partial-fractions]

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

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78 views

How would I find the integral $\int_0^1 \frac{x^2+x+2}{(x+1)(1+x^2)}\,\mathrm{d}x$?

The goal is to find this integral: $$\int_0^1\frac{x^2+x+2}{(x+1)(1+x^2)}\,\mathrm{d}x$$ I have been on this question for a long time and I am halfway through solving it: As shown above, using ...
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Decomposition of $\frac{10(s+6)}{\left[(s+3)^2+25\right](s^2+25)}$

Determine $\alpha,\beta,\gamma,\delta$ in $$\frac{10(s+6)}{\left[(s+3)^2+25\right](s^2+25)}=\frac{\alpha s+\beta}{(s+3)^2+25}+\frac{\gamma s+\delta}{s^2+25}$$ I have come across the partial fraction ...
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continued fraction formula . pls help

I'm self studying this book "Methods of Solving Number theory Problems by Elina" since many days but currently stuck on this formula of continued fractions. For example $a=87/ 55 = [1,1,1,...
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Let $a$ be a non zero real number. Evaluate the integral $\int \frac{-7x}{x^{4}-a^{4}}dx$

I hit a wall on this question. Below are my steps $$\int \frac{-7x}{x^{4}-a^{4}}dx=-7\int \frac{x}{x^{4}-a^{4}}dx$$ Let $u=\frac{x^2}{2}, dx = \frac{du}{x}, x^{4}=4u^{2}.$ $$-7\int \frac{1}{4u^{2}-a^{...
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Obtain the coefficient of $x^2$ in the expansion of $1+\frac{6}{2x+1}+\frac{5}{2-3x}$

Hello so this is a 2 part question and I managed to express that praction as a partial fraction which was equaled to $$1+\frac{6}{2x+1}+\frac{5}{2-3x}$$ I will add my work below I tried lot to Obtain ...
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How to find the $n^{th}$ derivative of $\dfrac{x^2-1}{(x-1)(x-2)(x-3)}$ [closed]

I am confused with the numerator part what to do after finding the partial fraction of the denominator please help. I have an exam coming up in a week
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I need help with a stupid webassign problem. It's about partial fraction

The question is following: Write out the form of the partial fraction decomposition of the function (as in this example). Do not determine the numerical values of the coefficients. $$\frac{1}{x^2+x^4}$...
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1answer
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Partial Integration doubt [closed]

I want to integrate an equation but I am unable to understand how do I convert the d(1-c) term to some dx term. Please help me understand such manipulations and share any helpful resources to ...
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36 views

Partial Fraction Decomposition methods

I have this problem where I have to solve the following integral. ${\displaystyle\int}\dfrac{1}{\left(x+a^2\right)\left(x+b\right)^2}\,\mathrm{d}x$ I checked the solution on https://www.integral-...
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Are there any shortcuts for computing the coefficients of partial fractions that are not covered by Cover Up Rule?

I am referring to powers of linear factors higher than 1, and all quadratic factors. I was just wondering if there are any obscure techniques to solve the coefficients since in common practice I know ...
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How to do the PFE of a function whose polynomial does not easily expand?

I want to do a PFE of $$ y(x) = \frac{1}{ x^2 + \sqrt{2}x +1} $$ when I try to expand the polynomial I end up with $$ y(x) = \frac{1}{(x + \sqrt{\frac{1}{2}})( x + \sqrt{\frac{1}{2}}) + \frac{1}{2}} $$...
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Infinite series with factorial only in the denominator using partial fractions [duplicate]

I am having trouble manipulating this and decomposing it to its partial fractions. We are asked to find the sum of this but I have tried to find references on summations with factorials in the ...
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59 views

Differential equation with partial fraction

How do you separate this differential equation into a partial fraction? Solve the following differential equation: $$\frac{dy}{dx}=\frac{2y^2-xy+x^2}{xy-x^2}$$
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Why does this partial fraction decomposition work, even with division by $0$?

Here is how partial fraction decomposition works. First, take a fraction like $\frac{1}{n(n+1)}$. You can express this fraction as the sum $\frac{A}n + \frac{B}{n+1}$ for some constants $A$ and $B$. ...
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How to integrate the logistic equation with partial fractions?

I’m trying to integrate the logistic equation, but I’m stuck with the partial fractions bit. None of the solutions I’ve found online (including here) clarify how it should go, and I keep getting ...
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54 views

Compute $\sum_{n=1}^\infty (\frac34)^n \frac{7n+32}{n(n+2)}$

Question: Compute $$\sum_{n=1}^\infty \left(\frac34\right)^n \frac{7n+32}{n(n+2)}$$ I first did the partial fraction decomposition into: $$\sum_{n=1}^\infty \left(\frac34\right)^n \frac{16}n - \sum_{...
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How to determine h(x) in a polynomial partial fraction decompostion

Im susposed to do a partial fractional division; $$ \frac{-2x^2 + 8x - 9} {(x-1) (x-3)^2}$$ Now I used the formula and this is what I got; $$\frac {A}{(x-1)} + \frac{B}{x-3} + \frac{C}{(x-3)^2}$$ Now ...
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Partial fractions of $\frac{x^4}{(x^2-1)^3}$ (is it possible to find coefficients without expanding?)

I want to decompose the fraction $\frac{x^4}{(x^2-1)^3}$ : $$\frac{x^4}{(x^2-1)^3}=\frac A{x-1}+\frac B{(x-1)^2}+\frac C{(x-1)^3}+\frac D{x+1}+\frac E{(x+1)^2}+\frac F{(x+1)^3}$$ Because $f(x)=\frac{x^...
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how to solve this into partial fractions

I'm having a bit of a hard time putting this into partial fractions: $$\frac{10}{x^2+2x+1+\pi^2}.$$ I know that the roots of the denominator are $-1 \pm i\pi$, but I dont know how to proceed on puting ...
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Finding nth derivative of $\frac{1}{x^4+4}$

I am supposed to find the nth order derivative of: $$\frac{1}{x^4+4}$$ I tried to resolve into partial fractions. But it didn't work out for me. Edit- where I am stuck $$\frac{1}{x^4+4}=\frac{1}{(x-1+...
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Approach ideas for the integral $\int\frac{dx}{(x^4-16)^2}$

Well, the title sums it up pretty well. I'm in search for some smart approach ideas for solving this indefinite integral: $$\int\frac{dx}{(x^4-16)^2}$$ I know one that would work for sure, namely ...
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When should I use “partial fraction decomposition” when integrating? [closed]

Does the difference in power have to be greater than 1? How do I know when to use u-substitution and when to use long division and when to use decomposition etc.? Thanks
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Intuition behind partial fractions

I was trying to solve the following integral: $$\int \frac{dx}{(x^2 + 2x + 1)(x^2 + 1)}$$ But I'm having some trouble doing partial fractions decomposition with this one. What I did was the following: ...
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General rule on partial fraction expansion?

I have fractions of the form $$F(p) = \prod_{i=1}^{n} \frac{1}{1+\alpha_i p} $$ and it appears that the partial fraction expansion is $$F(p) = \sum_{i=1}^{n} \frac{1}{(1+\alpha_i p)} \prod_{j=1,j\neq ...
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69 views

Partial fractions when numerator polynomial has greater degree than denominator

Consider the 'The Big example' as shown in this site, in it we are tasked to split: $$ \frac{x^2 +15}{(x^2+3)(x+3)^2} $$ They split it as: $$ \frac{A_1}{x+3} + \frac{A_2}{(x+3)^2} + \frac{Bx+C}{(x^2+3)...
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What does it mean to equate the coefficients of like terms when solving for A and B in partial fractions?

I'm trying to step myself through solving partial fractions in a year 10 book by Cambridge. This is a concept they're introducing early for students who want to challenge themselves and it's pretty ...
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81 views

Integral $\frac{2}{{(1+\tan x)}^2}$

Indefinite integral of $$ \int\frac{2}{(1+\tan x)^2} dx $$ I tried substituting $\tan x=t$, but I get to a partial fraction decomposition that I can't solve.
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Simplification of rational function

I saw the following simplification done: $$\frac{100}{\frac{900sL}{900+sL}+100}$$ is equal to $$\frac{sL + 900}{10sL + 900}$$ and was wondering how this is done? Thank you
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Partial fraction decomposition algorithm [duplicate]

So I was faced by this problem: If $p,q \in K[x]$, with $q = q_{0}^{\alpha_{0}} \dots q_{m}^{\alpha_{m}}$, all of $q_{i}$ irreducible and $\alpha_{i}$ natural numbers. So, there is some $p_{0} \dots ...
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Can't find the inverse of Laplace Transform.

Find $\mathcal{L}^{-1}{(F(s))}$, if given $F(s)=\dfrac{2}{s(s^2+4)}$. I have tried as below. To find inverse of Laplace transform, I want to make partial fraction as below. \begin{align*} \dfrac{2}{s(...
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1answer
24 views

Logistic differential population equations

Given a population growth $P$ milligrams at time $t$ hours, such that $\frac{dP}{dt} = 0.005P(120-P)$ If the initial population is $10$mg find $P$ as a function of $t$ in order to express how many ...
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Problem's solving the integral $\int_0^\infty \frac{1} {x^{1/3}(x+a)} dx$ where a is a constant

I found this integral $$ \int_{0}^{\infty} \frac{1} {x^{1/3}(x+a)} \, \mathrm{d}x $$ as part of an example of differentiation under the integral sign in the book Advanced Calculus Explored, by Hamza E....
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PFE with complex roots on TI-89

How to do partial fraction expansion with complex roots on a TI-89 (i.e. from formula #1 to its expansion form #2)? With real roots, I could use the Algebra/expand()...
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Verify Proof of Sum of Partial Fraction Coefficients

I've noticed that if you do partial fraction decomposition on $$f(x)=\frac{\prod(x-a_j)}{\prod(x-b_i)}=o\left(\frac1{x}\right),\ \text{as}\ x→∞,$$ (I hope I'm using the right "o" here), ...
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Calculate zeros of rational function directly from residues of partial fraction expansion without calculating polynomial coefficients

Imagine the following rational function in the Laplace-domain with $s = \mathrm{j}\omega$ $$G(s) = \sum_{i=1}^{m} \dfrac{c_i}{s-p_i}= h\,\dfrac{\prod\limits_{i=1}^{n}(s-z_i)}{\prod\limits_{i=1}^{m}(s-...
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Show the existence and evaluate $F'(t)$ knowing $F(t)$

We are given the following expression : $$F_1(t) = \int_{0}^{\pi}{\frac{\cos(x)dx}{(1+t\cos(x))^2}}$$ $$F_2(t) = \int_{0}^{1}{\frac{\log(1+tx)}{1+x^2}}dx$$ Show the existence and evaluate $F_1'(t)$ ...
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Decompose $\frac{1}{a \sin x +b}$ into sum of trigonometric functions

I am wondering if it is possible to write $$f(x)=\frac{1}{a \sin x +b}$$ as a sum of functions that don't have trigonometric functions in the denominator? Or just pure trigonometric functions, without ...
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Partial Fractions Repeated Term:

Second Answer to this question provided me a unique way to solve a problem involving partial fraction decomposition. A problem arose when I tried to solve this particular problem with a repeated term ...
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Quick Rejog of Partial Fraction Decomposition:

I don't need the problem to be solved I just need to have the decomposed equation. I have the following equation: \begin{equation} I=\int \frac{dx}{x^2(x^2-16)}\end{equation} The method that I have to ...
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140 views

Integral $\int\limits^{\infty}_0\frac{\tan^{-1}t }{(1+t)^{n+1}} dt$

I'm having a good amount of trouble evaluating this: $$\int\limits^{\infty}_0\frac{\tan^{-1}(t)dt}{(1+t)^{n+1}},\ n>0$$ Here are some methods I've tried: $$\int\limits^{\infty}_0\frac{\tan^{-1}(t)...
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1answer
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Simplification of complex rational?

I was looking at a problem in a textbook and found the following simplification: $$\frac{1}{x+i\omega} * \frac{2y}{y^2 + \omega^2} = \frac{2}{xy} * \frac{1}{1+i\frac{\omega}{x}}*\frac{1}{1 + (\frac{\...
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41 views

Decompose into simple fractions $\frac{f'}{f}$

Let $f(x) = (x-a_1)(x-a_2)...(x-a_n)$. Find a decomposition into simple fractions of $\frac{f'}{f}$. Where $f'$ is a derivative of our polynomial. As I understand, we have to find a pretty-format of $...
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Partial Differentiation $ \frac {log( \frac {S_0}{k}) + (r - \frac {\sigma^2}{2})T} {\sigma \sqrt{T}} $

I'm having difficulty in finding the $ \frac{\partial d1}{\partial \sigma}$ for the equation below. I tried using quotient rule but I'm getting $ -\sqrt{T} -\frac {log( \frac {S_0}{k}) + (r - \frac {\...
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48 views

Decompose into simple fractions over $\mathbb{C}$

Decompose into simple fractions over $\mathbb{C}$: $$f(x) = \frac{1}{(x^2-1)^n}$$ I know how to decompose some fractions by $\mathbb{R}$, but I have no idea how analyze this over $\mathbb{C}$ and ...
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1answer
38 views

Partial fraction method in Integration

My question is can we proceed with the method without even factorising the denominator. For eg take a question. $$\int \frac{2x-3}{(x^2-1)(2x+3)}dx$$ Can we write directly $$\frac{2x-3}{(x^2-1)(2x+3)}=...
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How would I approach Coefficient using partial fractions? [closed]

Find the Coefficient $c_k$ such that $\frac{(x)}{(1-x)^2}= \sum_{k=0}^\infty c_kx^k $ (Any hints)
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28 views

Find the finite power series of $\frac{u^3+3u^2+3u+1}{u^2+2u+3}$

I tried to do the partial fraction of $\frac{x^3}{(x-1)^4(x^2+2)}$. By letting $u=x-1$ I get $\frac{1}{u^4}\frac{u^3+3u^2+3u+1}{u^2+2u+3}$. I was able to get the answer by doing the long division of ...
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0answers
28 views

Using geometrical sum to infinity to do long division?

I was doing the partial fraction of $\frac{1}{x^6}(\frac{x-2}{x^2+1})$. By using long division I get: $$\frac{1}{x^6}(\frac{x-2}{x^2+1}) = \frac{1}{x^6}\left [ -2 + x + 2x^2 -x^3 -2x^4 + x^5+\frac{2x^...
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1answer
56 views

help reducing by modulo

I'm an engineer, not a math guy. Please use small words if possible. ;-) I am going through this neat paper on a method of partial fraction decomposition by repeated synthetic division. On page 157 ...
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21 views

partial fraction expansion with As+B and irreducible quadratic

My question is at the bottom. In an electrical engineering text on Laplace methods for solving electrical circuits a section on discussing p.f.e. the author says that when the denominator has an ...

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