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? [closed]
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:
$$\frac{1}{ar+1}-\...
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Decomposing a Fraction Involving Cube Roots for Integration
I had an exam the other day and there was this question to decide whether the following function is improperly integrable from 0 to 1. I wrote a solution for it but now I came to understand it's not ...
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Partial Fractions of $\frac{1}{(x+1)^{m}(x+3)^{n}}$
In the partial fractions decompostion of $$\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 ...
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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 ...
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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 ...
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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 ...
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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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