Questions tagged [partial-fractions]
Rewriting rational function in the form of partial fractions is often useful when calculating integrals.
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How to Evaluate the sum $\sum_{k=1}^n \frac{2 k+1}{k(k+1)} $ by the partial fractions $ \frac{1}{k}-\frac{1}{k+1} $ [closed]
I am trying to find the answer to this sum:
$$
\sum_{k=1}^n \frac{2 k+1}{k(k+1)}
$$
but I need to find it through a partial fraction method:
$$
\frac{1}{k}-\frac{1}{k+1}
$$
How di I do that? I also ...
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How does partial fraction expansion generalize to fractions of integers? Why is it not unique, in that case?
The Wikipedia page for partial fraction expansion mentions that it can be generalised to "regular" fractions, i.e. fractions of integers: https://en.wikipedia.org/wiki/...
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How do I complete this partial fraction
I am trying to take perform a partial fraction decomposition on the following function but something is not correct.$$Y(x)=\frac{(5x+10)(\cos x)}{x^2(x+9+i)(x+9-i)}$$ that I am trying to take the ...
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Partial fractions of equal degree [closed]
I have the integral and I thought I was supposed to use partial fraction decomposition, but I was not getting the correct answer.
$$\int\frac{x^3 + 3x^2 + 2x + 1}{x^3 + x}\mathrm dx$$
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Partial Fractions Decomposition-Unsure Which Method To Use When
So I was working on this problem and could not use the cover up method to solve it. I was getting the wrong answer.
Find B.
$$\frac{1}{s^2(s^2+4)}=\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+4}$$
$$s=0: ...
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Partial fraction decomposition with repeated using modulus and congruence
I'm trying to solve this exercise:
$$\Phi(x) = - \frac{1000x^5 + 6320x^4 + 13545x^3 + 12364x^2 + 5885x + 1017}{(2x + 5)^3 (5x + 2)^3} = \frac{h(x)}{(2x + 5)^3} + \frac{k(x)}{(5x + 2)^3}$$
Where $h(x)$ ...
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Partial Fraction Expansion Algebra Help
I hope someone can help. Given this equation
$$ F(s) = \frac{(1 - e^{-x})s^{-1}}{(1 - s^{-1})(1 - e^{-x}s^{-1})} $$
Apply a PFE
$$ = \frac{A_{1}}{1 - s^{-1}} + \frac{A_{2}}{1 - e^{-x}s^{-1}} $$
Then
$$...
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Imaginary numbers and partial fraction decomposition in integration
If I have a function whose denominator has only complex roots, could I integrate it and perform partial fraction decomposition by separating the factors? For example, am I allowed to evaluate $$\int\...
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Wy can a fraction with $(x+\alpha)^n$ in the denominator be partially decomposed into n different fractions?
My textbook in algebra states without proof that:
A rational function:
$$s(x)=\frac{p(x)}{(x-\alpha)^m(x-\beta)^n}$$
Where $\alpha \neq \beta$ and $\deg(p(x)<\deg(m+n)$, can always be
partially ...
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Help me to get deeper understanding of Euler's proof of his Arithmetical Theorem
With distinct numbers $a_1, a_2, \ldots, a_n$, let's denote the products of the differences of each of these numbers with the each of the rest of them by the following principle:
\begin{align}
(...
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Brachistochrone with friction (substitution and partial fractions)
This MathWorld article on the brachistochrone makes the following step in lines (29) and (30):
$$\left( 1 + y \prime ^ 2\right) \left( 1 + \mu y \prime \right) + 2 \left( y - \mu x \right) y \prime \...
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Simplify Using the Fundamental Theorem of Calculus
I am just starting my Partial Differential Equations course. I am looking at the steps to derive the heat equation for a rod, and there is one step I can't figure out. It says that the following can ...
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Solving the homogeneous first order ode $y' = \frac{2xy}{y^2-x^2}$
Solving the homogeneous first order ode $y' = \frac{2xy}{y^2-x^2}$
substituting $y=ux$ so that $y' = u + x\frac{du}{dx}$"
$u + x\frac{du}{dx} = \frac{2x^2u}{ux^2-x^2} = \frac{2u}{u^2-1}$
$\...
user637978
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Question: Fastest way to solve this? Can this be written as a matrix and solved via elimination? A, B, C, D and E are constants. x is arbitrary.
I would like to know if anyone has an idea what the best way to find the constants A, B, C, D, and E is. Can this be solved/written using a Matrix?
x can be chosen arbitrarily to find the constants A, ...
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Finding partial fractions including complex numbers
I got a similar question when finding the partial fraction decomposition.
Here $i$ is imaginary number.
I set up the equation like this and I am confused here:
$$
\frac{e^{ikx}}{(x-2i)(x+2i)}=\frac{...
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Stone-Weierstrass theorem and partial fraction decomposition
I'm reading notes about spectral theory and at some point it is stated that the $\mathbb{C}$-linear span of the functions $f_{\lambda}:\mathbb{R}\to\mathbb{C}:x\mapsto (x-\lambda)^{-1}$, where $\...
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Usual method of partial fractions decomposition over the reals seems to fail.
I assumed that it would be straightforward to find the partial fraction decomposition over the reals of the rational function $$f(x) = \frac{1}{(x^2 +1)^2}.$$ However, when I try what I thought would ...
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Show for |x|<1: [duplicate]
If |x|<1 show that:
$\sum_{k=1}^∞ kx^k = \frac{x}{(1-x)^2}$
I know that I should use partial fraction expansion. But don't really understand how to do it.
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Integrating rational functions with irreducible quadratic denominators; Which method to use and when?
I'm just getting back into math after a long absence and wanted to start from the beginning with some of my weaknesses during undergrad: in this case, partial fraction decomposition. The problem I ...
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What is this "trick" to solve for the unknowns in partial fractions?
For example, $\frac{4}{(x-2)(x+7)} = \frac{A}{x+7} + \frac{B}{x-2}$
setting the numerator: $4= A(x-2) + B(x+7) $
The method I always use is to expand and compare coefficients, which can be very long ...
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Question on partial fractions; why numerator has to be one lower degree than denominator?
When decomposing into a partial fraction, why does the highest degree of the numerator have to be one lower than the numerator?
For example:
$\frac{x}{x^3-1} = \frac{a}{x-1} + \frac{bx+c}{x^2+x+1}$
...
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Fourier series and partial fraction expansion of πcsc(πz)/z
Having trouble with this question involving the Fourier series and partial fraction expansion.
I need to prove
$$
z\sum_{n\in Z }{\frac{(-1)^n}{z^2-n^2} } = π\csc(πz)
$$
I think I should start by ...
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answer
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Tricky partial fractions
Im trying to do partial fraction but cant seem to get it right, and the examiner have not showed how he did the partial fraction, just the answer.
I want to partial fraction;
$$\frac{1}{((s+\frac{1}{2}...
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Proving limit $\lim_{x\to a_i}\Big(\left|\frac{q_i(x)}{(x-a_i)^{n_i}}\right|\Big)=\infty$ for partial fractions
From Vector Calculus, Linear Algebra and Differential Forms by John Hubbard, in the introductory Linear Algebra section, the author talks about how the dimension formula can be used to prove the ...
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Integration of $\int \frac{x^2}{(x-3)(x+2)^2}$ by partial fractions
Integration of $\int \frac{x^2}{(x-3)(x+2)^2}$
I know that we can use substitution to make the expression simplier before solving it, but I am trying to solve this by using partial fractions only.
$\...
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Inverse Laplace Transform of $\frac{\omega n^2}{s((s+z\omega n)^2+\omega n^2(1-z^2))}$
I've to solve the inverse Laplace transform of the following function, and am unable to find a good starting point. Any hints in the right direction would be appreciated. Thanks.
$$\mathcal{L}^{-1}\...
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Partial Fraction of $\frac{1-x^{11}}{(1-x)^4} $ for Generating Function
The original question involves using generating functions to solve for the number of integer solutions to the equation $c_1+c_2+c_3+c_4 = 20$ when $-3 \leq c_1, -3 \leq c_2, -5 \leq c_3 \leq 5, 0 \leq ...
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Hard Integration of rational fraction
For $j=0...n$ let be $a_j,b_j,c_j\in\mathbb{R}$ and $$B_j^n=\binom{n}{j}t^j(1-t)^{n-j}$$ the j-th Bernstein polynomial defined over the closed interval $[0,1]$.
Any hint on how to find a closed form ...
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How to develop $f(x)=-\frac{1}{(x+2)^2}$ into a power series?
I tried to develop the following
$f(x)=-\frac{1}{(x+2)^2}$
into a power series. However, I was not able to do that. Since I tried to split it into two fractions, and it didn't work neither for me or ...
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Is there a general strategy for computing these winding numbers?
In the question How to compute this winding number integral?, we have a special case of an integral of the form
\begin{equation}
I = \frac{1}{2\pi i}\int_{-\pi}^{\pi}\frac{H'(k)}{H(k)}dk,
\end{...
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Express in partial fractions and expand the terms using binomial expansion up to $x^3$ [closed]
$$
\frac{2}{(1-x)\left(1+x^{2}\right)}
$$
This is then split into partial fractions
$$
\frac{A}{1-x}+\frac{B x+C}{1+x^{2}}
$$
Computing this i had gotten
\begin{equation}
2=A\left(1+x^{2}\right)+(B x+...
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Is there a systematic method at use to turn something like $\frac{1}{2+u}\cdot\frac{1}{1+u^2}$ into $\frac{1}{2+u}-\frac{u-2}{1+u^2}$?
Is there some kind of method/trick/strategy/etc used to turn
$$\frac{1}{2+u}\cdot\frac{1}{1+u^2}$$
into
$$\frac{1}{5}\left [\frac{1}{2+u}-\frac{u-2}{1+u^2}\right ]$$
?
I run into such problems while ...
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Partial fraction decomposition for $\int \frac{-x^2+x+4}{(1-x)^2(3x+1)} dx$ [duplicate]
We want to evaluate the integral
$$\int \frac{-x^2+x+4}{(1-x)^2(3x+1)} dx$$
What I have troubles with, is to understand the principle of partial fraction decomposition.
For instance, here we'd have
$$\...
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Question regarding partial fraction decomposition for $\int \frac{1}{x^2-4} dx$
We want to integrate
$$\int \frac{1}{x^2-4} dx$$
I have tried the following:
$$\frac{1}{(x-2)(x+2)} = \frac{c_1}{x-2} + \frac{c_2}{x+2}$$
I wanted to find out the value of $c_1$ by multiplying with $(...
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answers
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Partial fraction decomposition of $(\frac{1}{n} + \frac{-1}{n+1})^p$
Context: I was trying to prove $\;\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^3(n+1)^3} = 10 - \pi^2$
$\displaystyle \frac{1}{n^3(n+1)^3} =
\left( \frac{1}{n} + \frac{-1}{n+1} \right) ^3 $
Partial ...
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Ordinary generating function - undo transformation without complex arithmetic
Ordinary generating function is very similar in concept to the $Z$ transform. For linear recurrences with constant coefficients we get rational functions. Suppose that we are limited to real partial ...
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A Laplace transform of a certain family of rational functions.
I was always interested in computing Laplace transforms. It was already during the course of my studies in the subject of electric circuits that I encountered this technique. Again, by using Laplace ...
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Integrate $\int\frac{3x}{x^5+x^4+1}dx$
I have an integral which I solved. But, I am not sure whether my answer is right or not. The integral is $$\int\frac{3x}{x^5+x^4+1}dx$$ My answer $$3\left(-\dfrac{\displaystyle\sum_{\left\{Z:\>Z^3-...
user1071701
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Inverse Laplace without Partial Fractions [closed]
How do I find Inverse Laplace of $s^3/(s^4+4a^4)$ without using Partial fractions. I solved it using Partial Fractions but I wonder if there is some way solving it using properties of Laplace ...
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Evaluating integral of the form $\int \frac {dx}{(x-b)^m(x-a)^n}$
$$\int \frac {dx}{(x-b)^3(x-a)^2}$$
Can someone please help evaluate this integral? Finding the partial fraction seems tedious and I don't know any other way.
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How to solve long integration by partial fraction decomposition problems faster?
Some problems are just too time consuming for short exam times what is the fastest way to solve problems like this one for example $$\int \frac{5x^4-21x^3+40x^2-37x+14}{(x-2)(x^2-2x+2)^2}dx$$
user1062191
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Calculate integral of $\frac{1}{z^4-1}$ using Cauchy's integral formula
I want to calculate the following integral
\begin{align*}
\int_{|z|=2}\frac{1}{z^4-1}dz
\end{align*}
using Cauchy's integral formula. As hint I have to use partial fraction decomposition to decompose $...
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Coefficients for partial fraction decomposition
I'm not sure if this question has been asked somewhere but I couldn't find an answer to it.
I need the coefficients in this partial fraction decomposition but in a specific way:$$\frac{1}{(x^2-b^2)^n}$...
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Why is that integral written as the product of 2 other integrals?
I am a student and just learning about integrals. **Could you please explain to me how the highlighted part of the equation was derived?** I can't quite understand how was the c/a term brought of the ...
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Why can we treat the numerator as a constant in partial fractions?
I'm learning the method of partial fraction decomposition as a 'useful dodge' (Silvanus Thompson, Calculus Made Easy) for calculus problems, but I'm not quite following the reasoning. According to ...
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Find particular solution for $(D^2+1)y=e^{a \cos x}$, where a is an arbitary constant.
I tried solving the problem as in the image by taking partial integrals and also by series expansion, but it is becoming more complex to continue and find a close form of it. Please solve it.
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Which integrating technique should I use?
Just some context: In the mathematical course, I have undertaken this year, I've just learnt how to integrate using partial fractions, substitution(not trig though, just a variable) and integrating ...
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Taking the partial fraction when the variable is exponential
I'm trying to solve the following integral:
$\int\frac{dx}{2^x+3}$
and here's what I've done so far:
Substituting $t=2^x$ we have $dt=2^x\ln 2 dx\implies \frac{dt}{t}=\ln2dx$
Now, I have to take the ...
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Integrate $2u/(u-u^3)$ [closed]
I'm currently trying to integrate:
$$
\int \! \frac{2u}{u-u^3} \, du = \ln \frac{u+1}{u-1} + \ln C
$$
I've tried to use partial fractions to simplify the
$$
\frac{1}{u-u^3} = \frac{1}{u} - \frac{1}{2 \...
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Integral of a partial fraction function$\int\frac{1}{(x-1)^3(x-2)^2}dx$
How do we determine the integral
$$\int\frac{1}{(x-1)^3(x-2)^2}dx$$
My ideas are the followings:
We just split them into partial fractions like $$\frac{1}{(x-1)^3(x-2)^2}=\frac{A}{x-1}+\frac{B}{(x-1)^...
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