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

learn more… | top users | synonyms

0
votes
3answers
48 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 $(t+5)(t-4)$. ...
0
votes
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) + ...
2
votes
0answers
45 views

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 ...
0
votes
1answer
17 views

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

${5x+4\over -x^2-x+2}$ My solution: ${5x+4\over -x^2-x+2}$ $= {5x+4\over -(x-1)(x+2)}$ $= {A\over (x-1)} + {B\over (x+2)}$ $= {3\over -(x-1)} + {2\over -(x+2)}$ $= -{3\over x-1} -{2\over x+2}$ ...
0
votes
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: $x+21 / 2(2x+3)(3x-2) = A / 2(2x+3) * B / 2(3x-2)$ $x+21 = A(2)(3x-2) + B(2x+3)$
0
votes
2answers
51 views

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.
1
vote
0answers
170 views

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) ]) ...
0
votes
1answer
32 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 : ...
0
votes
1answer
24 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} = ...
3
votes
1answer
121 views

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 ...
1
vote
2answers
143 views

$\int_0^\infty \frac{x^2}{(x^2-4)(x^2-9)}\,\text dx$

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 ...
0
votes
1answer
26 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} ...
1
vote
1answer
31 views

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 ...
1
vote
2answers
50 views

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 ...
0
votes
2answers
22 views

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 ...
0
votes
0answers
16 views

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 ...
0
votes
1answer
17 views

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 ...
1
vote
0answers
17 views

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 ...
1
vote
2answers
38 views

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 + ...
1
vote
3answers
45 views

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 ...
8
votes
1answer
115 views

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 ...
0
votes
0answers
53 views

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 ...
0
votes
0answers
38 views

Wolfram answer conflicts with mine partial fractions integrals

Wolfram computation in question My work
4
votes
3answers
163 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.
0
votes
1answer
74 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 ...
0
votes
2answers
34 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 ...
1
vote
0answers
23 views

DPE problem invlolving Fourier transforms / partial eq.

Don't even know where to start with this question! would really appreciate some guidance.
1
vote
0answers
40 views

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 ...
1
vote
2answers
46 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.
0
votes
1answer
32 views

Using partial fractions to integrate dy/(y(y-2))

This is a solution given for a practice exam I'm working through. However, I don't get where the 1/(-2)-(-0) (the part with the circled numbers in the denominator) came from. For instance, when I ...
1
vote
2answers
31 views

Expanding $\frac{1}{z^3-z}$

How would I use partial fractions to expand the following equation? $\frac{1}{z^3-z}$ I have tried changing re-writing that as: $\frac{1}{z(z-1)(z+1)}$ But I have a problem finding the numerators ...
1
vote
3answers
35 views

How do we decompose functions with fraction powers

Here is an example $\dfrac{1}{2+\sqrt x}$ I am tempted to use the form $\dfrac{Ax+C}{2+\sqrt x}$ Which will give me 1=Ax+C And if x=0 then C= 1, A= 0 This brings us to the same function we ...
0
votes
0answers
28 views

Integrate indefinite rational

Wolfram was of not help when solving this because the way I need to integrate these types of problems is by expressing the integrand as a sum of partial fractions. I won't get credit on the test if ...
1
vote
1answer
32 views

Partial decomposition of equations

Help me solve this $$\dfrac{x^3+x+2}{x(x^2+1)^2}$$ It looked like a simple one, but became complicated in my hands because i tried it like this: ...
2
votes
4answers
166 views

Find the sum $\sum_{n=1}^{50}\frac{1}{n^4+n^2+1}$

Find the sum $\sum_{n=1}^{50}\frac{1}{n^4+n^2+1}$ $$\begin{align}\frac{1}{n^4+n^2+1}& =\frac{1}{n^4+2n^2+1-n^2}\\ &=\frac{1}{(n^2+1)^2-n^2}\\ &=\frac{1}{(n^2+n+1)(n^2-n+1)}\\ ...
2
votes
1answer
27 views

Why do we have to add a term for every exponent when integrating by partial fractons?

For example, to integrate: $1/(1+x)^3$ you can write it as a partial fraction as $A/(1+x) + B/(1+x)^2 + C/(1+x)^3$ but i don't really understand why. I found that it is analogous to representing for ...
0
votes
2answers
75 views

Integral of $1/(x^2-1)^2$

I was having a bit of trouble with an integral $$\int {1 \over (x^2-1)^2}dx$$ It was given as a partial fractions problem, and I tried a substitution of ...
2
votes
1answer
86 views

Evaluate $\int{\frac{x^3}{x^3-3x^2+9x-27}}dx$

So for the equation below I used long division first, and got $\int{(1+\frac{-3x^2+9x-27}{x^3-3x^2+9x-27}})dx$ (So my signs should be the other way around, thank you to Alijah Ahmed for pointing it ...
0
votes
1answer
48 views

Partial fractions problem

How can I decompose this expression through partial fractions? $$\frac{15x^2−28x−72}{x^3−2x^2−24x}$$
3
votes
2answers
63 views

Evaluate $\int{\frac{\sqrt[3]{x+8}}{x}}dx$

So I've tried solving the equation below by using $u=x+8$, and I get $\int{\frac{\sqrt[3]{u}}{u-8}}du$ which doesn't seem to lead anywhere, I've also tried taking the $ln$ top and bottom, but I don't ...
2
votes
1answer
64 views

How to integrate $\int \frac{1}{(1-z^2)(1-z^n) }dz$?

In a text that I am reading: Let $\phi(z)=\frac{1}{2}\log\left(\frac{1+z}{1-z}\right),\omega (z)=z^{n},(n\in\mathbb{N}^{+})$, ...
0
votes
1answer
24 views

Euclidean Algorithm on $F[X]$

When we do partial fraction decomposition we recall that we can split off coprime factors in the denominator: $$ \frac{a(x)}{p(x)q(x)} = \frac{b(x)}{p(x)} + \frac{c(x)}{q(x)}, \,\,\,\text{where ...
1
vote
0answers
23 views

Determine lambda from a non-constant differentiable function of one variable

Suppose f is a non-constant differentiable function of one variable. Determine, with reasons, the value of $\lambda$ for which F(x, y) = f($\lambda x^{3}$ + y) satisfies the partial differential ...
0
votes
1answer
44 views

Ahlfors method for partial fraction decomposition

On page 31 in Complex Analysis by Ahlfors, he discusses decomposing a rational function into $$ R(z) = G(z) + H(z)\tag{12} $$ where $G(z)$ is a polynomial without a constant term and is the singular ...
1
vote
1answer
35 views

Partial Fraction Decomposition Clarification

I'm just looking for some overall clarification for the following cases. Now, to the extent of my knowledge, the following examples of partial fractions would be split up in the following way: ...
2
votes
1answer
57 views

solve $y'=\frac{2x+y-1}{x+y+2}$

I need some help solving this differential equation, I'm doing everything as I was taught but I just need help at the last step, I can't seem to get this right. the ode we want to solve is ...
1
vote
3answers
78 views

Is it possible to solve $\int \frac{1}{u^2-u}du$ without hyperbolic tangent?

I'm getting stuck on this integral, and all the tools I see online relate it to hyperbolic tangent. When I try to solve it I break it up using partial fraction decomposition to $\int ...
8
votes
2answers
70 views

How do I evaluate the integral $\int \frac{1}{x^3 -1}dx$

this is my first question here so I hope I did everything right. Still really new to LaTeX as well$$\int \frac{1}{x^3 -1}dx $$ I first used partial fractions to decompose this integral into two ...
0
votes
3answers
47 views

Partial fractions of $1/(z^2+2)$

How does one split $1/(z^2+2)$ into partial fractions? Normally I would factorise, but I cannot spot the solutions of $z^2+2=0$.
4
votes
4answers
80 views

Partial fractions and trig functions

A long time ago I wrote down a silly problem. It starts with Attempt to write $$\frac{1}{\sin(x)\cos(x)}$$ using partial fractions. and then goes on to prove a trig identity. I was wondering if ...