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

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Partial fraction decomposition of type $1/(x^2+k)$

I know that partial fraction of this can be written as: $$\frac{3x}{(1+x)(2+x)}=\frac{-3}{1+x}+\frac{6}{2+x}$$ Which can be done in these ways: ...
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How to integrate $\frac{x+4}{x^2+2x+5}$

Having a little trouble on how to break it up. How to integrate $$\frac{x+4}{x^2+2x+5}$$
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Inverse Laplace transform (using table) when denominator cannot be factored

Usually when performing inverse Laplace transforms, I decompose the function into partial fractions, and then look up standard transforms in a table. For example: $$Y(s) = ...
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33 views

Partial fractions where the denominator is one function

I need to solve this differential equation for x: $$ \frac{dv}{dx} = \frac{4000}{v} - 0.9v $$ Rearranging: $$ \frac{dx}{dv} = \frac{1}{4000v^{-1} - 0.9v} $$ How would I go about splitting this ...
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Decompose a fraction in a sum of two

Let's say that I have this fraction: $$ \frac{2x}{x^2+4x+3}$$ I would like to decompose in two fraction: $$ \frac{A}{x+3} + \frac{B}{x+1}$$ Which is the procedure for that? :)
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Partial fractions for $\pi \cot(\pi z)$

I want to derive $$\pi \cot(\pi z) = \sum_{-\infty}^{\infty}\frac{1}{z-n} + \frac{1}{n}$$ without taking derivatives. I know through Mittag Leffler that $$\pi \cot(\pi z) = g(z) ...
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Partial Fractions with an irreducible quadratic factor

$\int \frac{2}{(x-4)(x^2+2x+6)} dx$. this is a partial fraction with irreducible quadratic factors. I know how to set it up and I found A, B, and C. 2 = A((x^2)+2x+6) +(x-4)(Bx+C). then I plugged ...
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29 views

Using Laplace transforms to solve a convolution of two functions

Hi I have this problem where I need to take the convolution of functions and I am not sure if I got the right answer or something close so any advice or help would be very appreciated. So here is the ...
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Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: ...
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How to Decompose into Partial Fractions

Why is it useful to write: $$\frac{}{(x+1)^2(x-1)}=\frac{}{(x+1)^2}+\frac{}{x+1}+\frac{}{x-1}$$ and not: $$\frac{}{(x+1)^2(x-1)}=\frac{}{(x+1)^2}+\frac{}{x-1}$$ when decomposing into partial ...
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complex conjugate pairs of a quartic

I tried my hand at this question, which included finding the partial fractions of $\frac{x^2}{1-x^5}$. I found a factor of $1-x$ for the denominator, but I do not know how to work out the complex ...
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How to evaluate $\int \frac{x^6}{x^4-1} \, \mathrm{d}x.$

Evaluate the integral: $$\int \frac{x^6}{x^4-1} \, \mathrm{d}x$$ After a lot of help I have reached this point: $x^2 = Ax^3 - Ax + Bx^2 - B + Cx^3 + Cx^2 + Cx + C + Dx^3 - Dx^2 + Dx - D$ But now I ...
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1answer
22 views

Partial fraction that contain special function

In the following equation; $$ \frac{e^{\frac{(2c+5x)}{3x}} \mathop{E_{n}}\nolimits\!\left(x\right)}{(a+x)(b+x)} $$ 1- Can I apply the partial fraction to the above equation as the following: $$ ...
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How to simplify this mathematical expression?

I found this: Let's rewrite the integrand so that it's easier to integrate: $$\dfrac{x}{(x-2)^2(x+1)} = -\dfrac{1}{9x+9}+\dfrac{1}{9x-18}+\dfrac{2}{3(x-2)^2}$$ This is the mathematical ...
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1answer
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Integrating via partial fractions

My question is this: Why is it that fractions have to be split up in a very specific manner? For example if I have $\frac{5x}{(x+1)^2}$ this fraction HAS to be split up like this:$$\frac ...
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Integral of $\frac{x^2}{x^2-4}$

How would I integrate the following: $$\int \frac{x^2}{x^2-4}\ dx$$ We have covered three techniques for integration: substitution, integration by parts and partial fractions. I have tried partial ...
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How does Wolfram get from the first form to the second alternate form?

So, I was trying to compute an integral but I couldn't actually manage getting anywhere with it in its initial form. So, I inserted the function in Wolfram Alpha and I really got a nicer form (second ...
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Partial fraction (doubt)

I have this partial fraction $$\displaystyle\frac{1}{(2+x)^2(4+x)^2}$$ I tried to resolve using this method: ...
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1answer
51 views

Faster method for partial fractions

Is there a way to apply the "cover-up" method when solving for fractions of the following type? $$\frac {2x}{(x+1)(x^2+1)^2}$$ The long way would be; $$\frac {2x}{(x+1)(x^2+1)^2} = \frac {A}{x+1} + ...
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Help w/ Partial Fraction Decomposition

I need some help figuring out how to decompose $\displaystyle\frac{1}{x^4+1}$ into partial fractions. This is what I have done so far: $$\frac{1}{x^4+1} = \frac{1}{(x^2 - \sqrt{2}x + 1)(x^2 + ...
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Converting to a partial fraction.

I'm trying to do an inverse Laplace operation on $I(s)$ shown below but I'm struggling on finding what $A$ & $C$ are on the partial fraction and how to do it. I calculated what $B$ equals by ...
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59 views

Partial fractions on $(cx^2+dx+e)^n$

If I have $$\frac{ax+b}{(cx^2+dx+e)^n}$$ with real coefficients and $(cx^2+dx+e)$ has complex roots, what does $$\frac{ax+b}{[c(x-\alpha)(x-\alpha^*)]^n}$$ turn into, in terms of partial ...
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Partial Fractions Integration Question

$$\int\frac{x^5+x-1}{x^3 +1} dx$$ Have tried everything ... polynomial long division, partial fractions, trig substitution etc... Not for an assignment, so if a complete solution could be provided ...
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Partial fraction decomposition,how?

I need to decompose this fraction: $${x^2+1\over (x-1)^3\cdot(x+3)}$$ I tried to write it up like this: $${A\over (x-1)}+{B\over (x-1)^2}+{C\over (x-1)^3}+{D\over (x+3)}$$ But now i get ...
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Partial fraction integral

Question: $\int \dfrac{5 }{(x+1) (x^2 + 4) } dx $ Thought process: I'm treating it as a partial fraction since it certainly looks like one. I cannot seem to solve it besides looking at it in the ...
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Partial Fractions with a Repeated and a Irreducible Quadratic factor

I am trying to make this into a partial fractions form but i can't seem to find a way to do it. The question is here: Change into a partial fractions form. \begin{align} \frac{2s}{(s+1)^2(s^2 + ...
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28 views

Partial fraction expansion of generating functions (clarification of a proof)

I give below a part of Feller's. I am struggling to understand how equation 4.8 was derived. Any help will be much appreciated! Thanks
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39 views

Laplace transform of initial value problem, stuck on partial fractions.

The problem im given is: Use Laplace transforms to solve the initial value problem. $$\ddot x +x=\sin(2t)$$ $$x(0)=0=\dot x(0)$$ I first do the following Laplace transforms: $$\mathcal{L}\{\ ...
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1answer
55 views

inverse laplace using partial fractions and completing square

what is the inverse Laplace transform of this equation $$\frac{1}{(s+1)(s^2+s+1)}$$ I know that completing the square for the quadratic term is required to avoid complex roots and then I need to use ...
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1answer
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Partial Fraction Using Heaviside cover-up method

How to convert this equation into Partial fraction Using Heaviside Cover-up Method $$\frac{x^2}{(x+2)(2x+3)}$$ After trying to solve this I am ending up getting this which is incorrect : ...
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Contour Integrals with Partial Fractions

Sorry, I'm new at posting here, so forgive me for any mistakes I make. I'm trying to evaluate the following using a contour integral. I don't know how to use the Residue stuff yet, so I basically ...
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Problems expanding a function

I'm trying to expand the function f(x). Can someone please tell me what I'm doing wrong? Thanks! I started off with this function: $$ f(x) = {z^3 \over (z-{1 \over 4})(z-{3 \over 4})(z-{1 \over 2})} ...
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Laurent Expansion partial fractions

I have a function: $\frac{1+2z}{z^3 + z^2}$ for $0 < |z| < 1$ (about $z=0$) I need to find the Laurent expansion of this function. However, I'm a bit confused how to find the partial fractions ...
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1answer
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express $\frac{42}{ s^2 + 7s}$ as a partial fraction

express this rational function as a partial fraction $$ \frac{42}{s^{2} + 7s}$$ So i factories $$ \frac{42}{s(s+7)} = \frac{A}{s} + \frac{B}{s +7} $$ $$42 = A(s+7) + Bs$$ let s equal $-7$ ...
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Why a linear numerator for fractions with irreducible denominators?

For example: (2x^3+5x+1)/((x^2+4)(x^2+x+2)) breaks down to (ax+b/(x^2+4))+(cx+d/(x^2+x+2)). I have been told that since the denominators are irreducible, the numerators will be either linear or ...
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Partial fraction decomposition of a complicated rational function

Find the partial fraction decomposition of the rational function $\displaystyle \frac{2x^3+7x+5}{(x^2+x+2)(x^2+1)}$ I have tried dividing first but keep running into problem after problem, please ...
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Irreducible quadratic factors; partial fraction decomposition.

Please help me understand why there is Dx+E, Fx+G etc, instead of the regular A's, B's, C's etc. What is it about the irreducible quadratic in the denominator that makes it different on top?
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$\ln(y-3) = \ln(3-y)$? How?…

I am working on the same question as this one someone asked on yahoo answers... https://uk.answers.yahoo.com/question/index?qid=20101207140137AAK5c1v $$\int^2_1\frac{4y^2 - 7y - 12}{y(y+2)(y-3)} \, ...
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Why is this true with partial fractions?

Suppose you have a fraction like $\frac{x^2+2x}{x(x-2)^2}$. You can rewrite that as $$\frac{x^2+2x}{x(x-2)^2}=\frac{A}{x}+\frac{B}{x-2}+\frac{C}{(x-2)^2}.$$ Why is it that you must put the linear ...
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57 views

Partial fraction decomposition of a rational function

The form of the partial fraction decomposition of a rational function is given below. $$\frac{x−3x^2−26}{(x+1)(x^2+9)} = \frac{A}{x+1}+ \frac{Bx+C}{x^2+9}$$ What are the values of $A,B$ and $C$? ...
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Using a reverse polynomial for a partial fraction decomposition in a recurrence relation problem

I recently asked this question about finding the formula for: $$gn=g_{n−1}+g_{n−2}+n, g_0=1, g_1=2$$ On that question, I was able to get help to the point of generating this partial fraction ...
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Computing $\int_{|z|=2} {dz \over z^2 + 1}$

Goal: To compute $$ \int_{|z|=2} {dz \over z^2 + 1} $$ by decomposition of the integrand in partial fractions. Attempt: Let $\gamma$ be the circle around the origin of radius $2$. Let us ...
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138 views

Solve second order differential equation with Heaviside function using Laplace transform

The equation is: $$y'' + 3y = u_4(t)\cos(5(t-4)), \quad y(0) = 0, \quad y'(0) = -2$$ Here $u_4$ is the Heaviside function with activation switch at $t=4$. I can get all the way to the partial ...
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obtaining inverse z-transform by different methods

how can i obtain the z-transform of $X(z) = \frac{z+1}{(z-1)(z+2)^2}$ by: 1) Partial fraction expansion, 2) residue theorem, and 3) direct division method any help is appreciated.
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Show that f(x)=1-x+5x^2

$6$. Let $$f(x) =\frac{9x^2 + 4}{(2x + 1)(x − 2)^2}$$ (i) Express $f(x)$ in partial fractions. (ii) Show that, when $x$ is sufficiently small for $x^3$ and higher powers to be neglected, $$f(x) = 1 ...
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Integrating partial fractions

I have $\int{\frac{2x+1}{x^2+4x+4}}dx$ Factorising the denominator I have $\int{\frac{2x+1}{(x+2)(x+2)}}dx$ From there I split the top term into two parts to make it easier to integrate ...
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the partial fraction of ${\pi}\cot{\pi}z$ from the partial fraction of $\frac{\pi^{2}}{\sin^{2}{\pi}z}$

I want to deduce the equation $${\pi}\cot{\pi}z=\frac{1}{z}+ \sum_{n=1}^{\infty} \frac{2z}{z^{2}-n^{2}}$$ where the convergence is uniform on compact subsets of $\mathbb{C}-\mathbb{Z}$ from the ...
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2answers
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Patrial Fraction definite integral with non-real part

I have a question to find the area bounded by $y = \dfrac{x^2-4x-4}{x^2-4x-5}$ and the x-axis. First I found the bounds by solving where the numerator would equal zero. My result is $2\pm2\sqrt2$ so ...
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1answer
18 views

Partial fraction integral still results with irreducible quadratic

I have an integral with an irreducible quadratic. I'm supposed to use partial fractions to solve the integral. The original integral: $\int\dfrac{x^2}{(x-1)(x^2+4x+5)}dx$. I used partial fractions ...
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2answers
66 views

Partial Fractions - $\frac{x^3}{x^2 + 12x +36}$

Ok, so I know that since the numerator has a higher power that long division is needed. So after doing that, the main fraction is $\frac{-6x-36}{x^2 + 12x + 36}$. I think that's right. But my problem ...