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

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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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2answers
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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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1answer
28 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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2answers
102 views

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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Fractional differential equations

I need to know what the fractional derivative models in differentfield for example physical, electricity models and what is the gain or aim .For example we have an ODE and we play or change the ...
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1answer
21 views

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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Evaluate the integral: $\int \frac{x^6}{x^4-1} \, \mathrm{d}x$ [duplicate]

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

How to evaluate the following integral? $\int \frac{x^6}{x^4-1} \, \mathrm{d}x.$

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
50 views

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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4answers
120 views

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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2answers
50 views

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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5answers
66 views

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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2answers
40 views

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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1answer
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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2answers
47 views

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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3answers
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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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3answers
70 views

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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4answers
58 views

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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1answer
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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35 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
52 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
25 views

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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1answer
35 views

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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2answers
35 views

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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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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5answers
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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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1answer
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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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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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2answers
130 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
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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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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 ...
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55 views

Rationalising the Surds

Please help me rationalise and simplify: $$ \frac{1}{\sqrt[3]{2} - 1} \ - \ \frac{2}{\sqrt{3} - 2} \ . $$ I have tried using the cube of the denominator and the square of the denominator on the ...
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Partial fraction expansion of $\frac{1}{x(x+1)(x+2)\cdots(x+n)}$

I try to find a partial fraction expansion of $\dfrac{1}{\prod_{k=0}^n (x+k)}$ (to calculate its integral). After checking some values of $n$, I noticed that it seems to be true that ...
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2answers
43 views

Quadratic Partial Fraction Decomposition

I am trying to find the inverse laplace transform of $(s^2+4) \over (s-2)(s+2)$. The solution is $ {2\over(s-2)} - {2\over(s+2)} + 1 $. But I can't figure out how to break it up so I can find the ...
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2answers
49 views

Solving a partial fractions

I have set up partial fractions so that$$Aln^3x-B(x+x^2)=1-x^2$$ and $$ Cln^3x+D(x+x^2)=1+x^2$$ to set up and solve the following $$\alpha(1+x)+ \gamma x= A+C$$ and from $$\frac {D lnx-B}{ln^3x} ...