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

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Decomposition into partial fractions to compute an integral

I'm having problems with: $$\int_{-\infty}^{\infty}\frac{x^4+1}{x^6+1}dx$$ I was thinking: $\frac{x^4+1}{x^6+1}$ is an even function and the interval $(-\infty,\infty)$ is symmetric about 0, we ...
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partial fraction decomposition braindead

decompose $\frac{x^2-2x+3}{(x-1)^2(x^2+4)}$ the way my teacher wants us to solve is by substitution values for x, I set it up like this: (after setting the variables to the common denominator and ...
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1answer
22 views

Inversion of $z$-transform using partial fraction decomposition

I want to inverse a $z$-transform of this general form $$X(z) = \frac{b_0 + b_1z^{-1}+\cdots+b_Mz^{-M}}{a_0 + a_1z^{-1}+\cdots+a_Nz^{-N}}$$ where $M$ < $N$. In order to do this, I use partial ...
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1answer
24 views

Question in regard to solving for inverse laplace transform

I am having some confusion when it comes to solving for the inverse laplace transform. ( We are allowed the tables with the common values by the way). Il give an example. Take, ...
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4answers
369 views

How to solve certain types of integrals

I'm asking for a walk through of integrals in the form: $$\int \frac{a(x)}{b(x)}\,dx$$ where both $a(x)$ and $b(x)$ are polynomials in their lowest terms. For instance $$\int ...
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Expressing in terms of partial fractions

Simplify $$\frac{7x^3+52x^2+97x+60}{x^4+8x^3+15x^2-4x-20}$$ into the form $$\frac{A}{x+p}+\frac{B}{(x+p)^2}+\frac{C}{x+q}$$ where $A,B,C,p,q$ are constants. Thank you for any help given. I have been ...
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1answer
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Solving $x+2y+3z=100$ in nonnegative integers. [closed]

Solving for number of solution in set of non-negative integer of $$x+2y+3z=100$$ by generating function but finding problem in writing partial fraction of ...
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Getting Partial Fraction Decomposition Wrong

So I have done this solution: But it's wrong according to technology. I fail to see any error I've made in this solution, do you guys have any idea what's wrong?
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1answer
20 views

Show that the following expansion is valid for first $n$ terms

The question is as follows, I've gone ahead and attempted as much as I could of it twice, however the final answer that I'm receiving does not match. I'm omitting the partial fractions steps; if ...
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1answer
25 views

Simplification imaginary fractions

In an exercise, a partial fraction expansion has to be done. I have no problem with that, but one of the last steps includes a simplification as follows: \begin{equation*} \left( -\frac 12 - \frac 16 ...
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3answers
70 views

Evaluate the integral $\int \frac{dx}{x^3 + 2x^2 + 2x}$ of a rational function

Evaluate $$\int \frac{dx}{x^3 + 2x^2 + 2x}.$$ I have no idea how to approach this. I know how to solve rational functions with numerator as highest degree polynomial using division and remainder. ...
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1answer
30 views

Values tried for partial fraction decomposition

I'll explain my question with the following example from wikipedia. Suppose, we have a function: $$ f(x)=\frac{1}{x^2+2x-3} $$ Here, the denominator splits into two distinct linear factors: $$ ...
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1answer
27 views

Partial fraction question with squared

How could one apply partial fraction decomposition to $$\left(\frac{s}{s^2+4}\right)^2$$ I tried to separate by doing $$\frac{A}{s^2+4} + \frac{B}{s^2+4}$$ and I got strange solution $A+B=0$ and ...
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1answer
9 views

partial fractions specific question

so i thought i knew about partial fractions, but apparently i don't. i have the answer to a partial fraction but i can't figure out how you get to that answer. the value is $$ X(z) = Z*(Z+2)/(Z-1)^2 ...
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1answer
33 views

Proof using partial fractions

I have to prove this formula: $$\int \! \frac{1}{(x^2+\beta ^2)^{k+1}} \, \mathrm{d}x=\frac{1}{2k\beta ^2}\frac{x}{(x^2 +\beta^2)^k}+\frac{2k-1}{2k\beta^2}\int \! \frac{1}{(x^2+\beta ^2)^k} \, ...
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2answers
182 views

Problem with partial fractions

I have this integral: \begin{equation*} \int \! \frac{x-1}{(x+1)(x^2+9)} \, \mathrm{d}x. \end{equation*} I already split the denominator into two factors. Now, when I do partial fraction ...
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7answers
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How can I solve the improper integral $\int_{1}^\infty {dx \over {(x+1)(x+2)}}$

$$\int_{1}^\infty {dx \over {(x+1)(x+2)}}$$ I have the indefinite integral solved for: $$\ln(x+1)-\ln(x+2) + C$$ But I don't know how to finish with $[1, \infty]$.
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How can I solve the integral $ \int {1 \over {x(x+1)(x-2)}}dx$ using partial fractions?

$$ \int {1 \over {x(x+1)(x-2)}}dx$$ $$ \int {A \over x}+{B \over x+1}+{C \over x-2}dx $$ I then simplified out and got: $$1= x^2(A+B+C) +x(C-2B-A) -2A$$ $$A+B+C=0$$ $$C-2B-A=0$$ $$A=-{1 \over 2}$$ ...
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3answers
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How do I know when to use partial fractions or long divison with this integral? $ \int {{x^4+1} \over {x(x^2+1)^2}} dx$

$$ \int {{x^4+1} \over {x(x^2+1)^2}} dx$$ Is there a method to determine which way is better?
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1answer
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Having Trouble finding a simplified power series representation.

Partial fractions seemed the most efficient route to take. However, I am having trouble at the end.
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1answer
203 views

Partial fractions - different results when done in steps than not

We have: $\frac 1 {(1-x)(1+x)(1-2x)}$ If I do the partial fractions straight: $\frac 1 {(1-x)(1+x)(1-2x)}= \frac a {1-x} + \frac b {1+x} + \frac c {1-2x}$ I get: $a=-\frac 12, b = \frac 1 6, c=\frac ...
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50 views

I having trouble using partial fractions.

For the integral below I have to use partial fractions, however I am at a lost on how to do so. $$\int\frac{dt}{t^2-t-20}$$ The farthest I have gotten to is factoring the denominator to $(t+5)(t-4)$. ...
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2answers
18 views

Equating Coefficients in Partial fractions

I'm having a hard time figuring out A, B, and C for this problem. $$ \frac{8}{y^{3} - 4y} $$ All I've got so far is $$ \frac{A}{y} + \frac{B}{(y+2)} + \frac{C}{(y-2)} $$ $$ 8 = A(y+2)(y-2) + ...
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Series for $\sin(z) / \sin(\pi z)$

${\sin(z) \over \sin(\pi z)} = 1/\pi + {z \over \pi} \sum_{n \in {\bf Z} \setminus \{0\}} {(-1)^n \sin(n) \over n(z-n)}$ First I apply Mittag-Leffler's theorem, to see that the RHS is a mereomorphic ...
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1answer
19 views

Decomposing partial fraction with a denominator negative quadratic expression $-x^2$

${5x+4\over -x^2-x+2}$ My solution: ${5x+4\over -x^2-x+2}$ $= {5x+4\over -(x-1)(x+2)}$ $= {A\over (x-1)} + {B\over (x+2)}$ $= {3\over -(x-1)} + {2\over -(x+2)}$ $= -{3\over x-1} -{2\over x+2}$ ...
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1answer
22 views

Finding the partial fraction of $x+21/2(2x+3)(3x-2)$

$x+21 / 2(2x+3)(3x-2)$ The 2 in the denominator represents a problem for me: $x+21 / 2(2x+3)(3x-2) = A / 2(2x+3) * B / 2(3x-2)$ $x+21 = A(2)(3x-2) + B(2x+3)$
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Partial fractions expansion problem $\frac{x^3-1}{4x^3-x}$

I want to calculate integral of the fraction, but first how to find the partial fraction expansion of $\frac{x^3-1}{4x^3-x}$. How to expand denominator? I am a bit lost here.
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Global and local maxima in a weighted sum of logarithms of linear functionals?

Is is possible to describe, and locate efficiently, the maxima of the function below in the parameters $\mathbf{x}$ $$\sum_{i} p_i \log( N + \sum_j x_j[B_j +(A_j-B_j)\delta_{ij} + min(A_j,B_j) ]) ...
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1answer
33 views

Recursive formula for partial fraction decomposition of a specific kind of fractions

I need to make a partial fraction decomposition of the following fraction : $$ \frac{1}{(x-a)^2(x-b)^2(x-c)^2(x-d)^2(x-e)} $$ The problem is that Wolfram doesn't give any answer : ...
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1answer
28 views

Partial Fractions and Complex Integral

I have $\int_{C}\frac{e^z}{z^2 + a^2}$ where $a>0$ and $C$ is a positively oriented simple closed contour containing the circle $|z|=1$. I start with $$\frac{1}{z^2 + a^2} = ...
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1answer
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Where to find an algorithm for decomposing rational functions into elementary fractions?

Specifically I need to decompose $\frac1{(1-x)(1-x^n)^2}$ into $\frac{f(x)}{(1-x)^3}+\frac{g(x)}{1-x^n\vphantom{()^2}}+\frac{h(x)}{(1-x^n)^2}$ where $f(x)$, $g(x)$, $h(x)$ are polynomials. Surely ...
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$\int_0^\infty \frac{x^2}{(x^2-4)(x^2-9)}\,\text dx$

I am trying to compute the following contour integration but am quite stuck I have to evaluate it analytically, by extending it to the complex plane and solving an appropriate integral involving a ...
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1answer
27 views

How to Break Down Stairs

I have a fraction and I want to break down stairs. $$ F(z) = \frac{1 - 1.37 z^{-1} + 37 z^{-2}}{1 - z^{-1} + 0.6 z^{-2}} $$ I want it in form of $$ F(z) = A_{0} + \frac{1}{B_{1} z + \frac{1}{A_{1} ...
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1answer
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Complex partial fractions

Could anyone help me separate this into partial fractions: $$\frac{\cos(z)}{z^2+1}$$ where $z=x+iy$. I've factored the denominator to get $$\frac{\cos(z)}{(z+i)(z-i)}$$ but I'm not really sure where ...
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Making it possible to do a Fourier transform on it: $\frac{1}{(k+w)^2(a^2 +w^2)}$

Sorry for all the edits, I'm very stressed and not so used to Latex. Full question: consider a filter with impulse response $$h(t)=e^{-bt} u(t)$$ where $u$ is the unit step function. The input ...
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Partial Fractions as Power Series

I have the partial fraction sum $$f(i\omega)= a_0 + \frac{a_1}{\lambda_1+i\omega} + \frac{a_2}{\lambda_2+i\omega}$$ Which I want to represent as a power series in $ x = i\omega $ I thought that the ...
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Separate terms of different orders from fractional polynomial

I have an expression: $\frac{1}{1-A}+\frac{-12A^4D^3 + 4A^4D^2 -16A^3D^2+4A^3D -6A^2D - AD}{- 12A^4D^3 + 4A^4D^2+12A^3D^3 -20A^3D^2 +4A^3D +16A^2D^2 -11A^2D +A^2 +7AD -2A + 1}$ How do I write it as ...
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Second Order Approximation for a Polynomial

if I have an expression: $L=\frac{12a^3d^3-4wa^3d^2+16a^2d^2-4wa^2d+6ad+1}{12a^3d^3-4wa^3d^2-4a^2wd+16a^2d^2+7ad-aw+1}$ what is the second order approximation in $\frac{d}{w}$? I know that ...
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Extracting a function of a variable from an expression

I have this expression: $\frac{d+2wd}{2w+3wd-3d-w^2-1}$ Is there anyway I can write it just as a function of f(d)? [To me this looks like it is already a function of d, but I want to confirm if ...
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Integration of fraction with square root

I have a problem with integrating of fraction $$ \int \frac{x}{x^2 + 7 + \sqrt{x^2 + 7}} $$ I have tried to rewrite it as $\int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{x^4 + 13x^2 + 42} = \int \frac{x^3 + ...
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Partial Fractions- Is there a quicker way?

So with the fraction $$ (2s^2+5s+7)/(s+1)^3 $$ Is there a quicker way to solve this rather than equating the coefficients? I cant use the 'cover-up' method because its just one fraction This is ...
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Partial fraction expansion for non-rational functions

With regard to this answer to an inverse Laplace transform question, I derived the following result: $$\frac1{i 2 \pi} \int_{c-i \infty}^{c+i \infty} ds \, e^{s t} \Gamma(s)^2 = 2 K_0 \left ( 2 ...
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Solve for inverse Laplace transform using non-repeating complex partial fractions. (5.7-4)

Synopsis: Please check my work. I do not have a text "answers to odd problems" for reference as this is an "even" numbered problem. The following documents in good detail the steps taken to solve for ...
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Wolfram answer conflicts with mine partial fractions integrals

Wolfram computation in question My work
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167 views

Integral fraction of polynomials

I have this problem: $$\int \frac{-2x^2+6x+8}{x^4-4x+3}dx$$ I have tried using partial fractions, but I can't get solution. Thank you for any advice.
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1answer
80 views

Solve $a_{n+1} - a_n = n^2$ using generating functions

The Full Question Using the method of generating functions, solve $a_{n+1} - a_n = n^2$ where $a_0 = 1$ My Research Scanned the website for similar answers, reviewed the following links: Solve the ...
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40 views

Partial Fraction Decomposition for Laplace Transform

As part of trying to solve a differential equation using Laplace transforms, I have the fraction $\frac{-10s}{(s^2+2)(s^2+1)}$ which I am trying to perform partial fraction decomposition on so that I ...
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DPE problem invlolving Fourier transforms / partial eq.

Don't even know where to start with this question! would really appreciate some guidance.
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Integral using Partial Fraction Decomposition [solved]

I've been having a problem with this integral; although it seems I have the correct answer, It continues to be marked incorrect on an automated grading program. Can someone tell me where I'm going ...
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2answers
63 views

Separating addition terms in denominator

If I have a fraction such as: $\frac{1+d(6-4a)}{1-a+d(7-4a)}$ then how can I separate it so I have it as $\frac{1}{1-a}+(some-term)$ Thanks.