# 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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### 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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### 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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### 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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### 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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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, $\kappa=c-\Big(\... 0 votes 2 answers 69 views ### Why are these two integrals different even though they should be equal?$\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 ... 0 votes 1 answer 78 views ### 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 ... 1 vote 0 answers 21 views ### How was this partial fraction solved with 2 variables? How was this partial fraction decomposition done? partial fraction image 1 vote 1 answer 66 views ### 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 ... 0 votes 2 answers 26 views ### 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) + ... 2 votes 2 answers 87 views ### 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 ... -1 votes 2 answers 55 views ### 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 ... 0 votes 1 answer 104 views ### 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 ... 