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

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rewrite with partial fraction decomposition

I want to rewrite this statement: $$f(x)=\frac{x}{-x^2-x+1}$$ into this statement $$f(x)=\frac{\frac{1}{\sqrt5}}{1-(x \cdot \phi)}+\frac{-\frac{1}{\sqrt5}}{1-(x \cdot (1-\phi))}$$ provided $\phi = \...
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Partial fractions and using values not in domain

I'm studying partial fraction decomposition of rational expression. In this video the guy decompose this rational expression: $$ \frac{3x-8}{x^2-4x-5}$$ this becomes: $$\frac{3x-8}{(x-5)(x+1)} = \...
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38 views

Laurent expansion - Faster technique

I'm currently preparing for an exam in complex analysis. There is a type of exercise, where I need to compute Laurent expansions about different places. However, my ...
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how to partial fraction $\frac{1}{(x+1)^2}$

I need to integrate $\frac{1}{(x^2+2x+1)}$, so I need to use partial fraction as the polynomial can be factored as $\frac{1}{(x+1)^2}$. This is what I've tried: $$\frac{A}{(x+1)} + \frac{B}{(x+1)^2}$$...
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Deriving the sequence if the generating function is irreducible?

I am trying to better understand generating functions and how they can be derived / manipulated / etc. Right now I am operating on this identity, slightly modified from the answer here: For a ...
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How did Euler prove the partial fraction expansion of the cotangent function?

As far as we know, Euler was the first to prove $$ \pi \cot(\pi z) = \frac{1}{z} + \sum_{k=1}^\infty \left( \frac{1}{z-k} + \frac{1}{z+k} \right).$$ I've seen several modern proofs of it and they all ...
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Partial fraction integration problem

I'm trying to solve this integral by partial fraction: $$ \int \frac{2x-6} {(x-2)^2(x^2+4)} dx \ $$ i think i have to write the expression like $$ 2\int \frac{x-3} {(x-2)^3(x+2)} dx \ $$ Then i ...
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Partial fraction decomposition of $\pi\cdot \tan(\pi z)$

Evaluate the partial fraction decomposition of $\pi \tan(\pi z)$ $$2\pi \tan(\pi z)=\cot\left(\frac{\pi}{2}-\pi z\right)-\cot\left(\frac{\pi}{2}+\pi z\right)$$ $$=\frac{2}{1-2z}+\sum_{k=1}^\infty \...
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given an equation, find A and B

I can easily solve this problem by finding A and B, and then A+B. My question is where there is a way to obtain A+B without finding A and B first. The problem is supposed to be challenging, but it ...
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Why is (x-xi)^n still a linear factor (Partial Fraction Decomposition)?

When we perform a Partial Fraction Decomposition and one of the solutions of the denominator is a multiple solution (let's say quadratic), we write: $$\frac{A_{1}}{(x-x_{i})} + \frac{A_{2}}{(x-x_{i})^...
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Partial Fractions with Quadratic Factor

I understand that if we have a quadratic factor such as in $\frac{8}{(x^2 + 1)(2x-3)} $ and we want to decompose, we should have a linear factor above $ x^2 +1$. Is the reason behind this ...
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Integrate $ \int \frac{4x^2+2x}{(3x-1)(x^2+1)}dx$.

$$ \int \frac{4x^2+2x}{(3x-1)(x^2+1)}dx$$ I am having difficulty determining which integration technique to use for the above question. I have tried partial fraction decomposition with two different ...
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If $ \int_{0}^{2}\frac{ax+b}{(x^2+5x+6)^2}dx = \frac{7}{30}\;,$ Then value of $a^2+b^2$

If the value of Definite Integral $\displaystyle \int_{0}^{2}\frac{ax+b}{(x^2+5x+6)^2}dx = \frac{7}{30}\;,$ Then value of $a^2+b^2$ $\bf{My\; Try::}$ We can write it as $$\frac{1}{2}\int_{0}^{2}\...
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Why did this incorrect partial fraction decomposition produce the correct answer?

I was reviewing a classmate (call him Bob)'s work on an integration of a rational expression (although integration is involved, it's beyond the scope of this question). The problem was: $$\int\frac{...
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Laplace Transform - Partial Fractions [closed]

Please see attached image. I keep coming up with a irrational/complex coeffecient which is correct. Can you please help me put it partial fractions please Thanks
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Understanding Why Partial Fractions Works [duplicate]

I was wondering why, not how, partial fractions work the way we are normally taught to do. To be specific: We are told that, when we have a second degree expression on the bottom that can't be ...
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Show that every infinite recursive set is the range of a nondecreasing unbounded recursive function of one variable.

I came across this problem and I was not able to solve it: Show that every in finite recursive set is the range of a nondecreasing unbounded recursive function of one variable. Also, what would be ...
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1answer
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Partial Fraction Decomposition - Multiple Answers-Question

Now I do understand how partial fraction decomposition works and why you can do it, but there is one case that I don´t understand. And that is, the following: $$\frac{A_{1}}{(x-x_{1})} + \frac{A_{2}}{(...
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3answers
28 views

Complicated partial fraction expansion

I'm reading the book generatingfunctionology by Herbert Wilf and I came across a partial fraction expansion on page 20 that I cannot understand. The derivation is as follows: $$ \frac{1}{(1-x)(1-2x).....
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Partial fractions - alternative result

I have the following fraction: $$ \frac{z}{(z-1)(z-2)} $$ When I try to decompose it to partial fractions, I get: $$ -\frac{1}{z-1} + \frac{2}{z-2} $$ But the result in my book is: $$ -\frac{z}{z-...
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Partial Fraction Decomposition with complex poles

I have a function which I'd like to perform partial fraction decomposition on, to allow easier inverse laplace transform. $$F(s) = \frac{1}{s(s^2+140s+10^4)}$$ I begin with finding the poles $$s = ...
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integral of $\frac {\sqrt {x+4}}x$

So I'm trying to compute $\int\frac {\sqrt {x+4}}xdx$ . They tell me to use a substitution to make it a rational function. I set $u=\sqrt{x+4}$ and got the integrand $\frac {2u^2}{u^2-4} du$. With ...
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Partial fraction decompostion

Solve the partial fraction. Starting out with... $${x^2+1\over x^3-1}={x^2+1\over (x-1)(x^2+x+1)}$$ Then the partial fraction formula part of $\displaystyle {A\over x-1}+{Bx+C\over x^2+x+1}$. ...
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Evaluating the rational integral $\int \frac{x^2+3}{x^6(x^2+1)}dx $

Evaluate $$\int \frac{x^2+3}{x^6(x^2+1)}dx .$$ I am unable to break into partial fractions so I don't think it is the way to go. Neither is $x=\tan \theta$ substitution. Please give some hints. ...
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Splitting fractions with a linear denominator: $\frac{2x-1}{x+2}$

How can $$\frac{2x-1}{x+2}$$ be split to give $$A-\frac{B}{x+2}$$ where $A$ and $B$ are integers? The solution is $$2-\frac{5}{x+2}.$$
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integral problem $\int \frac{2 \lambda a}{\mathbf{ (e^{at}-1)\lambda \sigma^2+2ae^{-at}}}dt $

Does anybody know how to tackle the below integral? I am analyzing a formula derivation where this appears as the final calculation, but I don't know how to get it solved $$\int \frac{2 \lambda a}{\...
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2answers
28 views

winding number partial fractions

I need to compute the winding numbers for a particular contour integration (which I understand and is not relevant for this question). However, before doing so I need to use partial fractions to get $...
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How can I make sense of this basic algebraic manipulation backwards?

$$\frac{1}{n^2-1} = \frac12\left(\frac1{n-1}-\frac1{n+1}\right)$$ It is easy for me to understand this algebraic from right to left ←, but I struggle to find a reasonable way to work from left to ...
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Evaluate the integral of : $\int \:\frac{e^{2x}}{1+e^{3x}}dx$

Evaluate the integral of : $\int \:\frac{e^{2x}}{1+e^{3x}}dx$ So I start by making : $e^{2x}=t$ and this gives me the following : $\frac{1}{2}\int \:\frac{dt}{1+t\sqrt{t}}$ Then using symbolab ...
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Integrate algebraic fraction with constant on top?

I understand that if you have $\int \frac{1}{x + 1} dx$ you simply do $\ln(x + 1) + C$. Now I'm slight confused because in my text book, $\int \frac{31}{x - 4} dx$ evaluates to $31\ln(x - 4)$ but $\...
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Easy partial fraction decomposition with complex numbers

There is an easy method to perform a partial fraction decomposition - described here, under the "Repeated Real Roots" title, for the coefficient A2. The problem is ...
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How to compute $\int \frac{1}{(x^2+1)^2}dx$?

Suppose we know $\int \frac{1-x^2}{(x^2+1)^2}=\frac{x}{x^2+1}+C$ How to compute $\int \frac{1}{(x^2+1)^2}dx$? I tried writing it as $\frac{1+x^2-x^2}{(x^2+1)^2}=\frac{1-x^2}{(x^2+1)^2}+\frac{x^2}{(x^...
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How do you use partial fraction decomposition to break up $1/(s+4)^2$?

How do you use partial fraction decomposition to break up $1/(s+4)^2$? The usual method isn't giving me an answer.
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Question about partial fractions and the order of two linear factors

I have a question on the topic of partial fraction and decomposition. Lets say I have the integral of $\frac{1}{(x^2 - x - 2)}$. When I want to find out the A and B values, I first have to get linear ...
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1answer
35 views

Use Laplace Transformations to solve $y''+2y'+5y=3e^{-x}sin(x)$, with $y(0)=0$, $y'(0)=3$

I've gotten this far and I cannot proceed: $L[y]=\frac{L[3e^{-x}sin(x)]+3}{p^2+2p+5}= \frac{3}{((p+1)^2+1)(p^2+2p+5)}+\frac{3}{p^2+2p+5}$ I'm finding it impossible to find the inverse to solve for $...
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64 views

Sum of series (telescoping)

I have the following problem: $$\sum^{\infty}_{n=1}\dfrac{2}{\left(n+2\right)\sqrt{n}+n\sqrt{n+2}}$$ I should find the sum of this sequence. I tried to simplify but it does not work.
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Show $\frac{\pi}{\cos \pi z}=4\sum\limits_{n=0}^\infty \frac{(-1)^n(2n+1)}{(2n+1)^2-4z^2}$ using partial fraction decomposition of $\cot$ and $\tan$

I am asked to show that $\frac{\pi}{\cos \pi z}=4\sum\limits_{n=0}^\infty \frac{(-1)^n(2n+1)}{(2n+1)^2-4z^2}$. Here is what I have got so far, $\frac{\pi}{\cos (\pi z)}=\frac{\pi}{\sin(\pi(\frac{1}{2}-...
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(Laplace Method) $y'' - 4y' = 6e^{3t} - 3e^{-t}$

For this problem $y(0) = 1$ and $y'(0) = -1$ I need to solve this problem using this: \begin{align*} y(t) &\longrightarrow Y(s)\\ y'(t) &\longrightarrow sY(s) - y(0)\\ y''(t) &\...
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Which rules are used to make function like one in Laplace Transformations table?

I have function like this: $$\frac{s^2+3s+3}{(2s^2+7s+7)} $$ It needs to be brought to the level of Laplace Transformations from table, like these two: $$\frac{a}{(s-b)^2 + a^2} $$ $$\frac{s-b}{(s-b)...
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How to come from $\frac{z^{2}}{z^{2}+1}$ to $\frac{z/2}{z+i}+\frac{z/2}{z-i}$

As title says, how to come from $\frac{z^{2}}{z^{2}+1}$ to $\frac{z/2}{z+i}+\frac{z/2}{z-i}$? Here is what I did: $\frac{z^{2}}{z^{2}+1}=1-\frac{1}{z^{2}+1}=1-\frac{1}{(z-i)(z+i)}$
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Partial fraction of a generating function

I am solving a recurrence relation $a_0 = a_1 = a_2 = 1, a_{n+3} = a_{n+2} − 2a_{n+1} − 4a_n$ for $n \ge 0$ I got a generating function for this sequence $f(x) = \frac{2x^2+1}{4x^3+2x^2-x+1} $ ...
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How do we know the form of a partial fraction expansion?

If you take a quick gander at the table on this page: http://tutorial.math.lamar.edu/Classes/CalcII/PartialFractions.aspx You'll notice there are some rules about what the form of a partial fraction ...
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Can I plug imaginary unit to solve partial fraction?

for example, I have partial fraction to solve: $$ 1/(x^2+1)(x^2+4)=A/(x^2+1)+B/(x^2+4)\ $$ then I need to solve $$ 1=A(x^2+4)+B(x^2+1) $$ Can I plug x=i and 2i so that I can get the value of A and ...
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Use partial fractions to write $\frac{1}{x^2 + x – 1}$ as $\frac{A_1}{x – α_1} + \frac{A_2}{x – α_2}$ and write it as a power series

Find the roots $α_1$, $α_2$ of $x^2 + x – 1$ and use partial fractions to write $\frac{1}{x^2 + x – 1}$ as $\frac{A_1}{x – α_1} + \frac{A_2}{x – α_2}$ , for suitable $A_1, A_2$. Using the power series ...
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1answer
25 views

Why Bx+C for irreducible quadratic but not perfect square? (Partial fractions)

When you partial fraction something like $$\frac{x+2}{(x+3)(x+2)^2}$$ you make it $$\frac{a}{(x+3)}+\frac{b}{x+2} + \frac{c}{(x+2)^2}$$ but when you have $$\frac{10x^2+12x+20}{(x-2)(x^2+2x+4)}$$ you ...
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50 views

Integration Partial Fractions

I have the following problem: $$\int\left(\frac{x+2}{x^2+x+1}\right)dx$$ I received this by simplification of another integral. But my question is how to procede from here. Is there a way to simplify ...
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3answers
102 views

Integral of $\frac{1}{x\sqrt{x-1}}$ using partial fractions.

Find $$\int\frac{1}{x\sqrt{x-1}}\ dx.$$ I have attempted to use the $A/x + B/\sqrt{x-1}$ method. That does not work. I have tried a substitution with $t= \sqrt{x-1}$ and $t^2 +1 = x$. Finding the $...
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1answer
50 views

$\int_0^5 \frac{dx}{x^2-x-2}$

I am having some difficulty with this problem. I am getting a finite answer but when I put the equation into wolfram alpha to check my answer it says that the integral does not converge.Here is what I ...
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How was this factoring of $1-2x-x^2$ achieved?

If I factor $1-2x-x^2$ using the quadratic formula I get $$x=\frac{2\pm \sqrt{4-4(-1)(1)}}{2(-1)}$$ $$x=\frac{2\pm \sqrt{8}}{-2}$$ $$x=-1 \pm \sqrt{2}$$ Let $\alpha = -1 +\sqrt{2}$ and $\beta=-1-\...
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4answers
83 views

Partial fraction decomposition for $\int\frac{3x^3+1}{x^3-x^2+x-1}dx$

I have been at this for hours and I don't know what I am doing wrong. It's partial fraction decomposition that I am doing but I just can't seem to get what I am supposed to $$\int\frac{3x^3+1}{x^3-x^...