Questions tagged [partial-fractions]

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

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Why does the degree of the numerator have to be 1 less than that of the denominator?

Pretty sure one suggested duplicate will be: Why do we take the degree of numerator 1 degree less than the denominator? Here the only one answer describes why the numerator having the same degree as ...
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Find particular solution for $(D^2+1)y=e^{a \cos x}$, where a is an arbitary constant.

I tried solving the problem as in the image by taking partial integrals and also by series expansion, but it is becoming more complex to continue and find a close form of it. Please solve it.
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Which integrating technique should I use?

Just some context: In the mathematical course, I have undertaken this year, I've just learnt how to integrate using partial fractions, substitution(not trig though, just a variable) and integrating ...
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Taking the partial fraction when the variable is exponential

I'm trying to solve the following integral: $\int\frac{dx}{2^x+3}$ and here's what I've done so far: Substituting $t=2^x$ we have $dt=2^x\ln 2 dx\implies \frac{dt}{t}=\ln2dx$ Now, I have to take the ...
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Integrate $2u/(u-u^3)$ [closed]

I'm currently trying to integrate: $$ \int \! \frac{2u}{u-u^3} \, du = \ln \frac{u+1}{u-1} + \ln C $$ I've tried to use partial fractions to simplify the $$ \frac{1}{u-u^3} = \frac{1}{u} - \frac{1}{2 \...
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Integral of a partial fraction function$\int\frac{1}{(x-1)^3(x-2)^2}dx$

How do we determine the integral $$\int\frac{1}{(x-1)^3(x-2)^2}dx$$ My ideas are the followings: We just split them into partial fractions like $$\frac{1}{(x-1)^3(x-2)^2}=\frac{A}{x-1}+\frac{B}{(x-1)^...
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Evaluate the sum: $\sum_{n=0}^{\infty}\frac{x^{n+2}}{(n+2)\ n!}$

How to evaluate the below sum? $$\sum_{n=0}^{\infty}\frac{x^{n+2}}{(n+2)\ n!}$$ I was trying to find the integral of $x\ e^x$ without using Integral By Parts. Here's what I got so far: $$\begin{...
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$\frac{1}{2n}+\frac{1}{n+1}-\frac{3}{2\left(n+2\right)}$ = $\frac{1}{2n}+\frac{3}{2\left(n+1\right)}-\frac{4n+5}{2\left(n+1\right)\left(n+2\right)}$ [closed]

How can one go from $\frac{1}{2n}+\frac{1}{n+1}-\frac{3}{2\left(n+2\right)}$ to $\frac{1}{2n}+\frac{3}{2\left(n+1\right)}-\frac{4n+5}{2\left(n+1\right)\left(n+2\right)}$ I'm really trying for an hour ...
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How to separate the following fraction into partial fractions?

I have that $$\frac{x^2(1-x)}{(1-x)^3} = \frac{x^2}{(1-x)^3} -\frac{x^3}{(1-x)^3} $$ Assuming we begin by cancelling the $(1-x)$ on top with one of those on the bottom, how do we go about splitting $\...
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Is there a symbol for comparing coefficients?

While doing PFs, ODEs or many other things, comparing coefficients often come up. Is it wrong to use the '=' for comparing coefficients, e.g. '$4x^2$ + 3x = Ax ∴ A=3'. Or is there a correct symbol to ...
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prove a partial fraction expansion formula

Let $f(x) = (x-a_1)(x-a_2)\cdots ( x-a_n), n\ge 1,$ where the $a_i$'s are distinct real numbers. For $k=0,1,\cdots, n-1$, prove that the partial fraction expansion of $\frac{x^k}{f(x)}$ is $\dfrac{x^k}...
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Solution to $\int\frac{\ln(1+x^2)}{x^2}dx$

Q: $\int\frac{\ln(1+x^2)}{x^2}dx$ Here is my entire working: So, overall, I started with the reverse product rule, then onto reverse chain rule and then tried to partial fraction, however, I still ...
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Finding the partial fractions decomposition of $\frac{9}{(1+2x)(2-x)^2} $

So this is basically my textbook work for my class, where we are practicing algebra with partial fractions. I understand the basics of decomposition, but I do not understand how to do it when then the ...
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Find minimum of the function using AM-GM

Problem: Find the minimum of the function $f(x,y)=x + \frac{8}{y(x-y)}$, where $x>y>0$ using AM-GM. My attempt: $$f(x,y)=2\cdot \frac{x+\frac{8}{y(x-y)}}{2} \ge 2 \sqrt{\frac{8x}{y(x-y)}}$$ But ...
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When should I use partial fractions in generating functions

I am currently studying generating functions and I don't understand why should I use partial fraction decomposition when solving $x^n$ coefficient of a question. For example, in this function $$\frac{...
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How to decompose $\frac{1}{(1 + x)(1 - x)^2}$ into partial fractions

Good Day. I was trying to decompose $$\frac{1}{(1 + x)(1 - x)^2}$$ into partial fractions. $$\frac{1}{(1 + x)(1 - x)^2} = \frac{A}{1 + x} + \frac{B}{(1 - x)^2}$$ $$1 = A(1 - x)^ 2 + B(1 + x)$$ ...
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Partial fraction with complex roots

Is it so that partial fractions with complex roots can work sometime, and sometime not? I have tried to check a result by WA here, and tried to solve it manually: \begin{equation} X(z)=\frac{104z+30}{...
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A problematic partial fraction decomposition $ X(z)=\frac{z}{(z-3)(z^2+4z+5)}$ [closed]

I try to solve this partial fraction: \begin{equation} X(z)=\frac{z}{(z-3)(z^2+4z+5)} \end{equation} and use the following form \begin{equation} X(z)=\frac{A}{(z-3)}+\frac{Bz+C}{(z^2+4z+5)} \end{...
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How to Solve This Integral With Multiple Variables? I think I should use partial fraction decomposition.

The problem is the integral of $$\int {\frac{-8 x}{x^4-a^4}}\, dx$$ I factored out the -8 and divided the x. I tried to use partial fraction decomposition but it wasn't forming into something I could ...
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Integral Using Partial Fraction Decomposition

So I have the integral of (4x^2+2x-1)/(x^3+x^2), and I have to solve it using partial fraction decomposition. The only thing is, the way I set it up, I need another factor for x to make it equal one ...
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Finding partial fractions of $\frac{z^3+2z^2-2z}{(z-2)(z^2+2}$ and/or using Cauchy's formula to solve

I am trying to find the inverse z-transform of \begin{equation} x(z)=\frac{z^3+2z^2-2z}{(z-2)(z^2+2)} \end{equation} and for this we need to get partial fractions. I have tried multiple approaches, ...
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Partial fractions and residue theorem

I need to find the inverse Laplace transform of this function: $$F(s) = \frac{50(s+1)}{s(s^2+20s+116)(0.8s+1)} $$ $$ \frac{50(s+1)}{s(s^2+20s+116)(0.8s+1)} = \frac{K_1}{s} + \frac{K_2}{0.8s+1} + \frac{...
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Integrating $\int \frac{1}{x\sqrt{3-x^2}}dx$ without trig sub

So I am evaluating $\int \frac{1}{x\sqrt{3-x^2}}dx$ without using trig sub integrals. So far I have $$u=\sqrt{3-x^2}, x^2=3-u^2,du=-\frac{x}{\sqrt{3-x^2}}dx, dx = -\frac{\sqrt{3-x^2}}{x}$$ So ...
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Integration by Partial Fraction Decomposition, Given Arbitrary Constants

Given Newton's 2nd law equation, I'm supposed to find $v$. My second law equation is: $m\dot{v}=-bv-vc^2$ By separation of variables, I arrive at $$\frac{dv}{bv+cv^2}=\frac{-1}{m}dt$$ $$\int_{v_0}^{v}...
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Question about specific step in proving Schur's Theorem (Combinatorics)

I refer to p.98 of generatingfunctionology in proving Schur's Theorem: The partial fraction expansion of $\mathcal{H}(x)$ is of the form \begin{align*} \mathcal{H}(x) &= \frac{1}{(1-x^{a_1})(1-x^{...
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How do we calculate the inverse Laplace transform of $F(s)=\frac{s^2+1}{(s+1)(s-1)}$?

We have \begin{equation} F(s)=\frac{s^2+1}{(s+1)(s-1)} \end{equation} which I want to use Heavisde method to find the fractions. We start \begin{equation} F(s)=\frac{s^2+1}{(s+1)(s-1)}=\frac{A}{(s+1)}+...
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Two partial fraction approaches, one is wrong, the other is right, why?

I want to do a partial fraction on \begin{equation} \frac{z}{(z-4)(z+\frac{1}{2})} \end{equation} Method one, which apparently is wrong: \begin{equation} \frac{z}{(z-4)(z+\frac{1}{2})}=\frac{A}{z-4}+\...
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Partial fraction decomposition involving imaginary numbers and two variables

I am trying to find a partial fraction decomposition for the following: $$\frac{1}{(-\alpha xi+4y)(\alpha xi + 2y)}$$ where $\alpha\in \mathbb{R}$. I am understanding that I could write this ...
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How to expand $\frac{1}{n^a(n+k)^b}$ using partial fraction decomposition?

Is it possible to decompose $\displaystyle\frac{1}{n^a(n+k)^b}$ into finite summation? where $a,b,n,k\in Z^{+}$ and $a+b$ is odd. What I tried is converting the fraction to double integral: $\...
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If $\int_{0}^{\infty}\frac{dx}{1+x^2+x^4}=\frac{\pi \sqrt{n}}{2n}$, then $n=$

If $\int_{0}^{\infty}\frac{dx}{1+x^2+x^4}=\frac{\pi \sqrt{n}}{2n}$, then $n=$ $\text{A) }1 \space \space \space \space \space\text{B) }2 \space \space \space \space \space\text{C) }3 \space \space \...
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Path to proving partial fractions and the fundamental theorem of algebra

As I've learned Calculus, I've tried to follow along with proofs of the rules that I use. In most cases, like say the Power Rule, I'm able to follow along with the proofs using concepts I understand, ...
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Solving this system for Partial Fraction Decomposition

I have a rational fraction $\frac{P(x)}{Q(x)}$ and would transform it into a sum of separate fractions. I know that $\{a_n\}$ is the set of the roots of $Q(x)$ which is of grade $t$, so it has exactly ...
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2 answers
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Evaluation of integral from textbook

Integral in question $$ \int\frac{dx}{\sqrt{\cos(x)}\sin(x)}$$ (If it helps, the original question in my textbook is to find the definite integral corresponding to this antiderivative with the limits ...
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Partial Fractions with Two Repeated Linear Terms

Rewrite the expression below into partial fractions $$\frac{\omega s}{(s+\omega)^2(s-\omega)^2}$$ I started by taking the general form $$\frac{A}{(s+\omega)} + \frac{B}{(s+\omega)^2} + \frac{C}{(s-\...
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Partial Fractions of $\frac{1}{x^6+1}$

I am trying to solve the following integral : $$\int \frac{1}{1+x^6}dx$$ I do not want the reader to evaluate the integral, but rather the partial fractions of the integrand : $\frac{1}{1+x^6}$. This ...
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Partial fraction of a function with two variables

I am trying to decompose a function within an optimization problem. I see that Maple and similar software products can do it for a single variable function but not for multi-variate ones. Now, I am ...
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How do I express this type of equation as partial fraction?

I got this equation by doing a Laplace transformation. Now, I want to find out the inverse Laplace and for that first I need to decompose this equation but I'm bit confused about how to express this ...
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A question on partial fractions

I have the given function which I must convert to partial fractions: \begin{equation} \frac{x^2}{(x^2+1)(x^2+9)} \end{equation} and I thought that I should prepare this as: \begin{equation} \frac{A}{(...
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Partial fraction decomposition trouble with a problem

I have this integral: $$ \int \frac{1}{(1+x^2)(1+(z-x)^2)} {\rm d}x $$ and I want to perform partial fraction decomposition in this form $$ \int \left( \frac{Ax + B}{1+x^2} + \frac{Cx + D }{1+(z-x)^2}...
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Are there applications of partial fraction decomposition ( of a rational function) outside integration problems?

I've been recently acquainted with a well known technique called " partial fraction decomposition" which allows, for example to express $\frac {x} {x^2-1}$ as $\frac {1}{2(x+1)} + \frac {1} ...
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How can I solve this primitive function?

The primitive function I'm trying to solve. $\int_\frac{1}{x^4-1}\;dx$ I've used partial fraction decomposition method. The following is the equation I'm setting up to be able to solve A, B and C:$(1/...
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To determine the integration of $ \int_{0}^{+\infty} \exp\!\Big(-\Big(\frac{ax^2+bx+c}{gx+h}\Big)\Big) dx$.

What is the integration of the following function: $$ \int_\nolimits{0}^{+\infty} \exp\!\bigg(-\bigg(\frac{ax^2+bx+c}{gx+h}\bigg) \bigg)dx.$$ What I have done is as follows: Here, $\kappa=c-\Big(\...
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Why are these two integrals different even though they should be equal?

$\int\frac{x^2}{x^2-4}dx$ and $\int\frac{x^2-4}{x^2-4}dx+\int\frac{4}{x^2-4}dx$ The first one is $\ln |x-2|-\ln|x+2|$ and the second one is $x+\frac{1}{4}\ln |x-2|-\frac{1}{4}\ln|x+2|$. Shouldn't they ...
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Partial Fraction Decomposition (Complex Numbers)

I'm going insane with this question from a previous exam: How do I get the partial fraction decomposition of: $${15 \over (z-3i)(2z-3)}$$ I don't understand how to 'equate' anything here. If we have ...
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How was this partial fraction solved with 2 variables?

How was this partial fraction decomposition done? partial fraction image
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Partial fraction decomposition done with square root. How is it possible?

I just stumbled on this example: $$\lim _{n\to \infty }\frac{\frac{\sqrt{n^3+n}}{n^4-n^2}}{\frac{n^{\frac{3}{2}}}{n^4}}=\lim _{n\to \infty }\frac{\sqrt{1+\frac{1}{n^2}}}{1-\frac{1}{n^2}}$$ And can't ...
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How to know if partial fractions have been done incorrectly?

Say you start with a set of fractions already broken up: $$ 2 + \frac{3}{x-1} + \frac{1}{x-3} $$ These can be combined into a single fraction by cross multiplying them: $$ \frac{2(x-1)(x-3) + 3(x-3) + ...
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2 votes
2 answers
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Strange/Unexpected behavior of an Infinite product

Some friends and I were playing around with this continued fraction: We noticed when writing it out for each next step, the end behavior went either to 1 (when there was an even number of terms) or ...
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Calculate partial fractions

calculate partial fractions for: $1/x^2(x^2 + 1)$ I have tried solving by expanding it like this: $A/x^2 + B/ (x^2 + 1)$ and it results in the right answer as given in class. But partial fractions ...
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Skepticism concerning Heaviside's "Cover-up Method" for $\textbf{partial fraction decomposition}$

I was reading this paper from MIT and it introduces Heaviside’s Cover-up Method for partial fraction decomposition. In that paper in Example $1$ it solves a problem using that method and just when ...
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