Questions tagged [partial-fractions]

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

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Partial fraction decomposition of $\int \frac{x^2+x+1}{x^2(x^2+1)^2}$ [closed]

I don't understand why the partial fractions of $ \frac{x^2+x+1}{x^2(x^2+1)^2}$ aren't $\frac{A}{x^2} + \frac{B}{x^2+1} + \frac{C}{(x^2+1)^2}$? Also, which one is the right answer? $$ =\frac{A}{x} + ...
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Prove the following integrals' equality

$$ \int_a^b \frac{x^2 - 2ax + a^2}{2x^2 - 2(a + b)x + a^2 + b^2} dx = \int_a^b \frac{x^2 - 2bx + b^2}{2x^2 - 2(a + b)x + a^2 + b^2} dx $$ I tried to solve this by Partial Fractions, and did $$ 2x^2 - ...
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Sum of finite series using partial fraction

I'm quite stuck with the following problem. I have seen on this forum that there is already an answer for the infinite sum to the problem but I can't seem to find how to find the sum for a finite ...
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Formula for $kth$ convergent of a simple continued fraction

Let $\frac{p}{k}$ be a rational number with partial fraction decomposition $C_n=[a_0; a_1,...a_n]$ and for $k<n$ let $C_k=[a_0;a_1,...a_k]$ be the partial fraction decomposition of the $kth$ ...
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How do I solve the partial fractions $ \frac{x+1}{(x^2+1)(x^2+x+1)}$? [closed]

I have the following problem: How can I break this fraction down into a simple fraction $ \frac{x^3+x^2+x+1}{(x^2+1)^2(x^2+x+1)}$ Can you help me? I applied the algorithm, but I couldn't solve it. ...
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27 views

Partial fractions of complex roots

I've just reviewed this answer: partial fraction for complex roots Whilst I understand most of it, there is one component I am having some trouble understanding. in particular $$\frac {2s^2+5s+12} {(...
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1answer
29 views

Decomposing an inverse Laplace transform

The given inverse Laplace transform is: $$\mathscr{L}^{-1}\left[\frac{5s^2+12s-4}{s^3-2s^2+4s-8} \right]$$ First split it up into three separate fractions and factorize the denominator $$\mathscr{L}^{...
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Partial Fraction Decomposition of $\int_{0}^{\infty} \frac{e^{-\frac{w}{s}}}{\left(mw+A\right) \left( mw+ B\right)^{L}}dw$

I'm sorry the title not allow more than 150 characters, I couldn't put a full-length integral equation I tried to simplify the equation to decrease less than 150 char. Here is the below full of the ...
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Irreducible quadratic partial fraction

Why is there Bx+c term when we try to split partial fraction with irreducible quadratic? Eg: $$\frac{1}{x(x^2+1)} = \frac{A}{x} + \frac{Bx+C}{x^2+1}$$ I think that splitting partial fraction is ...
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power series centered at origin of $\frac{1}{1 - z - 2z^2}$

I'm working on finding power series of $\frac{1}{1 - z - 2z^2}$. And I found post about it here and The conclusion seems fair enough: $$\frac{1}{1-z-2 z^2} = \sum_{k=0}^{\infty} a_k z^k$$ $$a_k = \...
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1answer
41 views

How to find Laurent-series?

I am trying to find the Laurent series for the function $\frac{1}{ z (2i - z)}$. I already obtained for... (1) ... $2 < | z |$: $\frac{1}{ z (2i - z)} = \frac{1}{2 i} \left( \frac{1}{z} + \frac{...
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Partial Fraction help [closed]

I wish to turn the following expression into a partial fraction $\ \dfrac{1}{N(1-\dfrac{N^2}{k})}$. For those interested, the expression refers to the per capita growth rate of a species.
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Why do we add extra coefficients for repeated terms when we do partial fraction decomposition?

I've seen how to do partial fraction decomposition but couldn't really see or understand why we add more coefficients. For example $$\frac{x^2+3}{(x-1)(x-2)^2} = \frac{A_1}{x-1} + \frac{B_1}{x-2} + \...
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Integral $\int \frac{2x^5-2x^4+2x^3+3}{2x^4-2x^3-x^2+1}dx$ as partial fraction solved using matrix equation

in order to solve the integral $$\int \frac{2x^5-2x^4+2x^3+3}{2x^4-2x^3-x^2+1}\mathrm dx,$$ the expression inside the integral can be expressed as $$(2x^5-2x^4+2x^3+3/2x^4-2x^3-x^2+1)= x+(A/(x-1))+(B/...
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27 views

How to do partial fraction decomposition from a Laplace transform

I am confused on how to do partial fraction decomposition from a Laplace transform. If you have $$\frac{d^2y}{dt^2} +3y = u_4(t)\cos(5(t-4)) \\ y(0)=0, y'(0)=-2$$ I can get to: $$L(y) = \dfrac{se^...
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116 views

Could $\int\frac{1}{x^{N+1}(x-1)}dx$ be solved analytically?

I am trying to solve this integral: $$\int\frac{1}{x^{N+1}(x-1)}dx$$ I have tried integration by partial fraction, substitution and by parts. But, I can't solve it. So, I would like to ask could this ...
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Nice integral $\int_{0}^{\infty}\ln\Big(\frac{x^3-x^2-x+1}{x^3+x^2+x+1}\Big)\frac{1}{x}dx=-\frac{3\pi^2}{4}$

Last integral of the day : $$\int_{0}^{\infty}\ln\Big(\frac{x^3-x^2-x+1}{x^3+x^2+x+1}\Big)\frac{1}{x}dx=-\frac{3\pi^2}{4}$$ I have tried integration by parts and some obvious substitution but I ...
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Let $f(x) = \frac{-2x+4}{(x^2+1)(x-1)^2}$. Express the function $f(x)$ as a sum of partial fractions.

I have gotten $(-2x+4) = A(x-1)^2 + B((x^2+1))$ But after letting $x = 1$, $B = 1$, I couldn't find $A$. Any help is appreciated. Thank you
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Partial Fractions $\frac{2}{s^2+4} = \frac{As+B}{s^2+4}$

I have this relatively simple partial fraction $$\frac{2}{s^2+4} = \frac{As+B}{s^2+4}$$ I multiply each side by $(s^2+4)$ and all that remains is $2 = As + B$. Then can I match the coefficients up ...
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How can I evaluate this integral by partial decomposition?

How can I evaluate this integral, if the denominator has a quadratic factors e.i-($b^{2}-4ac<0$) $$\int\frac{xdx}{x^{2}+6x+13}$$ by partial decomposition?
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Compute $\int\frac {x^2}{x^4+1}dx$ via partial fractions

I am trying to solve it with "partial fractions" $$\frac {x^2}{x^4+1}=\frac{x^2}{(x^2+x\sqrt{2}+1)(x^2-x\sqrt{2}+1)}=\frac{Ax+B}{(x^2+x\sqrt{2}+1)}+\frac{Cx+D}{(x^2-x\sqrt{2}+1)}$$ and I get the ...
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Row Echelon, Partial Fractions, and Numerator Coefficients

I am trying to get the numerator values for the partial fraction decomposition of: $$\dfrac{x^2+1}{x(x-1)(x+1)(x-4)}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1}+\frac{D}{x-4}$$ I really started hitting ...
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How the partial fraction decomposition works for finding this Inverse Laplace Transform?

I've been working to find inverse Laplace transform for the following : $$ \frac{A}{(s-a)(s-r_1)(s-r_2)} $$ However, I'm getting stuck on the partial fraction decomposition. When I run the ...
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How to get from $(Ae^t.t)/(t+1)^2$ to $Ae^t/1+t - Ae^t/(1+t)^2$?

I've been told that $(Ae^t.t)/(t+1)^2$ $=$ $Ae^t/1+t - Ae^t/(1+t)^2$ but im not sure how. I thought of using partial fractions but im not quite sure which case this would be. Would appreciate the ...
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Why is $\sum_{i=1}^n\frac n{n-i+1}=n\sum_{i=1}^n\frac1i$?

While reading through my (algorithms and) probability script, I have seen this equality for calculating the first moment of the coupon-collector problem. However, I don't quite see how the sum of the ...
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Partial fractions decomposition. Why $cx+d$ instead of $cx$ for the numerator of $(x^2+2$).

I understand that the aim of partial fractions decomp. is simply to reach (an) integrable functions, but then I have trouble wrapping my head around why you cannot make the numerator of something like ...
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general form of coefficients in partial fraction decomposition

Given $$ \frac{P(s)}{Q(s)} = \frac{A_{1}}{s-r_{1}} + \dots + \frac{A_n }{s-r_n},$$ where $P(s)$ is a polynomial with degree less than $n$ and $Q(s)$ is a polynomial with degree $n$ and with $r_1 ,\...
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Partial fraction decomposition involving implicit coefficient

This arised when I was attempting to solve an IVP using laplace transforms ; I'm not sure how to proceed when decomposing partial fractions involving implicit coefficient on the right hand side such ...
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1answer
49 views

Using Laplace transform to solve IVP involving complex roots

I am attempting to solve the following IVP : $$ y'' + y' + \frac54 y =t -(t-\pi /2 )u_{\pi /2}(t) \quad, y(0) =0 ,\quad y'(0) = 0. \tag{1}$$ My reasoning is as follows, \begin{align*} \mathcal{L}[y'' +...
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partial fraction - complex roots

i really can't understand how to manage this P(s) in order to apply inverse laplace transform $P(s) = \frac{2s-5}{1.5s^2-3s+4}$ i've tried this expansion: $P(s) = \frac{A}{s-1-1.29i} +\frac{B}{s-1+...
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Partial Frac Exp

Someone can explain step by step how i can manage this P(s) in order to apply the inverse laplace transform ? $ P(s) = \frac{s^3+5s^2+3}{s^2(s^2-3s-18)}$ I tried this way of fractioning but it ...
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70 views

If the value of integral in the image below is π then what is the value of y?

I could not simplify $$ \int_0^1 \sqrt{-1 + \sqrt{\frac{1+y}{x} - y}}\ dx $$ I tried integration it in an online integrator but trust me the result is seriously daunting to be back traced to $\pi$ as ...
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1answer
26 views

How to simplify series in a fraction

Given $$\\ A =\frac{1}{1 \cdot 2} + \frac{1}{3 \cdot 4} + \frac{1}{5 \cdot 6} + \ ... \ + \frac{1}{1997 \cdot 1998} \\ B =\frac{1}{1000 \cdot 1998} + \frac{1}{1001 \cdot 1997} + ... + \frac{1}{1998 \...
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80 views

Easy way to find the partial fraction

I always have trouble trying to find the partial fraction, especially for complicated ones. For example, this is what I will do to find the partial fraction of $\displaystyle \frac{8x^3+35x^2+42x+27}...
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Any shortcuts for integrating $\frac{x^6}{(x-2)^2(1-x)^5}$ by partial fractions?

Is there a faster way to get the partial fraction decomposition of this $\frac{x^6}{(x-2)^2(1-x)^5}$? $\frac{x^6}{(x-2)^2(1-x)^5} = \frac{A_1}{x-2} + \frac{A_2}{(x-2)^2} + \frac{B_1}{1-x} + \frac{B_2}...
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Are partial fraction decompositions unique

For a given rational function $f(x)=P(x)/Q(x)$ with degree of $Q(x)$ greater than degree of $P(x)$ are there more than one way to split it up into partial fractions? I do realize that the ...
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Splitting fraction in Fourier domain into two terms, where one is a derivative

I'm reading a solution to a problem that deals with solving a differential equation using the Fourier transform, and I can't follow one of the steps. In the Fourier domain we have: $$- {4 \over ({4 + \...
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1answer
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Finding the value of $\sum_{n=1}^\infty\frac{n^2}{(n+1)(n+2)(n+3)(n+4)}$ [duplicate]

Problem_ Find the value of $$\sum_{n=1}^\infty\frac{n^2}{(n+1)(n+2)(n+3)(n+4)}$$ It seems like I have to use the partial sum in order to get the exact value. But making it into the partial ...
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49 views

needs help to check slight difference result of integration

I check my answer but i found myself wrong. Here is the problem: $$\int_0^ {1.8} \frac{1}{\sqrt{x}(1+x)}dx$$. substitute $ u = \sqrt{x} ,\frac{dx}{\sqrt{x}}= 2du$ $$\int_0^{1.8} \frac{2du}{(1+u^2)}...
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Using partial fraction to power that less than 1

I need help to check this answer since, unfortunately, the worksheet doesn't attach the key. Integrate: $$\int \frac{1}{\sqrt{x} - \sqrt[3]{x}}dx$$ So, I tried use a substitution that allows the ...
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1answer
45 views

Tricks to compare complicated fractions?

Solving an integral I find: \begin{equation} I = \int \exp \left\{-N\left ( \epsilon r+\log(1+r) +\frac{rx^2(1-r\tau + \tau r^2 (\tau +1))}{1+r}+\frac{ry^2(-1+r\tau + \tau r^2 (\tau -1))}{1+r}\right)...
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Partial Fraction problem expansion dead end

I am having a problem solving this partial fraction, $$\frac{18+21x-x^2}{(x-5)(x+2)^2}$$ Which Solves to: $$\frac{2}{x-5}-\frac3{x+2}+\frac4{(x+2)^2}$$ However no matter how hard I try I always ...
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1answer
85 views

Prove that the rational function $f(x)/g(x)$ has a partial fraction decomposition in the case when $g(x)$ factors into distinct linear factors.

To be honest i don't even know where to start. I thought about using diagonalizable matrices and characteristic polynomials, but the class hasn't gotten there yet so there should be a way to solve ...
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1answer
97 views

Deriving the partial fraction decomposition of a hypergeometric function

I am studying a research paper and I don't understand how to derive a partial fraction representation of algebraic expressions whose image I am posting here. Démonstration On écrit $$R_n(t)(t+...
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2answers
51 views

What's the best way to solve $1 = A(x^2+1) + (Bx+C)(x+1)$

What's the best way to solve $$1 = A(x^2+1) + (Bx+C)(x+1)$$ I let $x=-1$ and got $A=\frac{1}{2}$ But what sub is ideal to find B&C This gets messy quick, I think. Instead, I started over, and I ...
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1answer
36 views

How do I perform this Partial Fraction Decomp.?

Disclaimer: I am not a student trying to get free internet homework help. I am an adult who is learning Calculus. I am deeply grateful to the members of this community for their time. $$\int{\frac{1}{...
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129 views

Infinity Summation problem (ANSWERED)

Show that $$\sum_{n=3}^∞ 96n^4+156n^3+33n^2+54n+3 = \frac{100-π^3}{87}$$ Only way I know how to solve this is to put it in my graphing calculator, but that isn't much help in this situation. This is ...
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4answers
94 views

integration of. $\int \frac{x}{x^3-3x+2}$

I am trying to integrate : $\Large \int \frac{x}{x^3-3x+2}dx$ I decomposed the fraction and got : $\Large \frac{x}{x^3-3x+2} = \frac {x}{(x-1)^2(x+2)}$ Then I tried to get two different ...
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1answer
22 views

Casting and Partial Fractions for symbolic generating function?

This is probably a silly question, but I can't seem to find an answer anywhere. It seems odd to me that Sage should allow us to get a series expansion for a generating function, but won't allow us to ...
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15 views

show that the portion of the partial fraction expansion of $P(x)/Q(x)$ corresponding to the factor $x-r$ Is $A/(x-r)$ where $A=P(r)/Q’(r)$

If $r$ Is a non-repeated root of $Q(x)$, show that the portion of the partial fraction expansion of $P(x)/Q(x)$ corresponding to the factor $x-r$ Is $A/(x-r)$ where $A=P(r)/Q’(r)$ I’m looking for ...

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