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Questions tagged [partial-fractions]

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

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1answer
34 views

Solution of $\left(D^2+1\right)y=\sin x$ by partial fractions and direct formula

To find the particular integral of the equation $\left(D^2+1\right)y=\sin x $,I converted this equation to its inverse $y=\frac{1}{D^2+1}\sin x$ . The solution to this can be obtained by taking the ...
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1answer
45 views

Finding General Formula for Coefficients of Partial Fractions

I am trying to evaluate the integral as written below and I've tried the following: $$\int\prod_{i=1}^{m}\dfrac{1}{x-i}\mathrm dx=\int\sum_{i=1}^{m}\dfrac{a_i}{x- i}\mathrm dx=\sum_{i=1}^{m}a_i\ln\...
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3answers
69 views

How is $\frac{2}{4u^{2} - 1}$ split up into 2 partial fractions? [closed]

Can somebody please explain to me the process of how the fraction $$\frac{2}{4u^{2} - 1}$$ is split up into the two fractions seen below? $$I = \int_{1}^{2}\frac{2}{4u^{2}-1} \,\mathrm{d}u = \int_{...
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4answers
766 views

How to quickly solve partial fractions equation?

Often I am dealing with an integral of let's say: $$\int\frac{dt}{(t-2)(t+3)}$$ or $$\int \frac{dt}{t(t-4)}$$ or to make this a more general case in which I am interested the most: $$\int \frac{...
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2answers
67 views

Find $f^{(22)}(0)$ for $f(x)=\frac{x}{x^{3}-x^{2}+2x-2}$

Find $f^{(22)}(0)$ for $f(x)=\frac{x}{x^{3}-x^{2}+2x-2}$ I know that I should use Taylor's theorem and create power series. However I don't have idea how I can find $a_{n}$ such that $f(x)=\sum_{n=1}^...
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6answers
76 views

Why partial fraction decomposition of $\frac{1}{s^2(s+2)}$ is $\frac{A}{s}+\frac{B}{s^2}+\frac{C}{(s+2)}$?

Can someone please explain why: $$\frac{1}{s^2(s+2)}=\frac{A}{s}+\frac{B}{s^2}+\frac{C}{(s+2)}$$ And not:$$\frac{1}{s^2(s+2)}=\frac{A}{s^2}+\frac{B}{(s+2)}$$ I'm a bit confused where the extra s ...
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2answers
32 views

Partial fraction decomposition of even function

I need to do partial fraction decomposition of this function (to solve its integral): $\frac{t^2}{t^4+4}$ Since $t^4+4=(t^2+2t+2)(t^2-2t+2)$ I would do: $\frac{t^2}{t^4+4}=A\frac{2t+2}{t^2+2t+2}+B\...
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1answer
33 views

Trivial indefinite integral with roots - can you spot the mistake?

Apparently my result is correct. Nevertheless, can someone take a look at the final expression and judge if the absolute values inside logarithm have been correctly reduced? Can anybody spot the ...
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1answer
33 views

How would I apply partial fraction expansion to this expression?

$$\displaystyle\frac{1}{X\bigg(1-\dfrac{X}{Y}\bigg)\bigg(\dfrac{X}{Z}-1\bigg)}$$ I want it in the form $$\frac{A}{X} + \frac{B}{(1-\dfrac{X}{Y})}+\frac{C}{(\dfrac{X}{Z}-1)}$$where I am required to ...
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2answers
56 views

Finding a faster method of partial fraction decomposition.

While integrating: $$ \int \cot^{-1} (x^2-x+1) dx $$ I had to do a large number of partial fraction decompositions, the following functions: $$ \frac {x(1-2x)}{(x^2-x+1)^2+1}, $$ $$ \frac {x}{(x^2+1)(...
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0answers
9 views

Inverse Laplace with fractional power

What is inverse laplace for $y=\{-16s^{v-1}+2s^{2v-1}/s^{2v}-4s^{v}+13\}$ where $v$ is fraction . The answer that I need to get after applying invers laplace is $y=[E(2t^v)][\cos(3t^v)-5\sin(3t^v)]$ ...
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4answers
876 views

Question about partial fractions with irreducible quadratic factors

Given this rational function: $$\frac{-4x^4-2x^3-26x^2-8x-44}{(x+1)(x^2 +3)^2}$$ The decomposition would look like this: $$\frac{A}{x+1} + \frac{Bx+C}{(x^2+3)} + \frac{Dx+E}{(x^2+3)^2}$$ And the ...
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1answer
22 views

How to rearrange this fraction so it matches a Laplace Transform table identity?

I have the fraction: $$\frac{s}{s^2+2s+2}$$ I want to rearrange the fraction so that I can solve find the inverse Laplace of it using the following identities from a Laplace Transform table: $$f(t)....
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1answer
52 views

Partial Fraction Decomposition. Need help doing u-substitution for a particular integral to solve. [closed]

$$\int\dfrac{3x^2+3x+1}{x^3+x}\mathrm dx$$ This is the work i currently have so far. Any tips would be appreciated to assist me with helping solve this equation.
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1answer
42 views

Is there a way to tell how to factor the denom. when doing a partial fraction?

My question comes from this specific problem from my homework: $$\frac{x}{81x^4 - 1}$$ Initially, I factored the denominator out to $(9x^2+1)(9x^2-1)$ and used this to find the $A,B,C,D$ to ...
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1answer
48 views

Evaluating indefinite integrals of the form $\int \frac{x^2 \,dx}{a x^5 + b}$

Evaluate the indefinite integral $$\int \frac{x^2 \,dx}{a x^5 + b},$$ for real parameters $a, b \neq 0$. No apparent substitutions simplify the expression (if the exponent of $x$ were an integral ...
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3answers
70 views

How can the correct form of the partial fractions decomposition be found for arbitrary rational functions?

What is the reasoning or intuition that leads to the assumption that $$r(x) =\frac{x^2 + 2}{ (x+2)(x-1)^2}$$ can be expressed as $$r(x) = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+2}$$ (For ...
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1answer
47 views

Use algebra to prove all proper rational function can be written in partial fraction decomposition

I'm learning the general tactics to integrate all rational functions and here's a fact that is written in the notes. It can be shown using algebra that every proper rational function $f$ can be ...
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0answers
32 views

Inverting a Laplace transform (for a Lévy process)

Let $\psi(\theta) = c\theta + \frac{\sigma^{2}}{2}\theta^{2} - \frac{\lambda\theta}{\alpha + \theta}.$ For those who are wondering where this function comes from, $\psi$ is the Laplace exponent for a ...
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5answers
78 views

Evaluate the indefinite integral $\int \frac{dt}{(t^2-1)^2}$ of a rational function

I have this problem: $$\int \frac{dt}{(t^2-1)^2}$$ and I'm a bit unsure about how to proceed. I could use partial fractions: $$\frac{1}{(t+1)(t-1)(t+1)(t-1)}$$ $$\frac{1}{(t+1)^2(t-1)^2}$$ $$\...
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3answers
44 views

Integration by partial fraction decomposition

I have the following example question in my homework and for the life of me I cannot understand the second to last step which I circled in red. Why are they suddenly changing the factors of 1/4 to 1/...
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1answer
55 views

Partial fraction decomposition. Why does the term appear multiple times when the linear term has an exponent? [duplicate]

I am reading this text: When there is a linear factor that appears r times, why do we have to write the factor up to r times in the factorization? So in the example under ...
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3answers
73 views

Using Laplace transform to solve differential equation $y'' -4y' = -4te^{2t}$

$y'' -4y' = -4te^{2t}, y(0)=0, y'(0)=1$ If you take laplace Transform of all terms, isolate L(y), I got $$L(y) = \frac{1}{(p-2)^2} + \frac{-2}{(p-2)^2 -4}$$ Then, taking inverse Laplace, you get $$y(...
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1answer
40 views

Confusion about “picking values of $x$”, partial fraction decomposition

One of the methods of decomposing a fraction and finding the constants, $A, B, C,$ etc., is to "pick values of $x$". For example, to find $A$ and $B$ of $${{3x}\over(x-1)(x+2)} = {A\over(x-1)} + {B\...
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2answers
142 views

The Integral $\int \frac {dx}{(x^2-2ax+b)^n}$

Recently I came across this general integral, $$\int \frac {dx}{(x^2-2ax+b)^n}$$ Putting $x^2-2ax+b=0$ we have, $$x = a±\sqrt {a^2-b} = a±\sqrt {∆}$$ Hence the integrand can be written as, $$ \frac {1}...
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4answers
98 views

How to do partial fraction decomposition with complex roots?

I want to determine $\int\frac{1}{x^2+x+1}\, dx$ which I approached by partial fraction decomposition. The complex roots of the denominator are $z_{1}=-0.5+i \frac{\sqrt 3}{2}$ and $z_2=-0.5-i\frac{\...
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2answers
37 views

What is the partial fraction of $\frac{x}{((x)^2+1)^2}$

I was trying to find the partial fraction of $$\frac{x}{(x^2+1)^2}$$ By the method of assuming $$\frac{x}{(x^2+1)^2}=\frac{(Ax+B)}{(x^2+1)} + \frac{(Cx+D)}{(x^2+1)^2} $$ But, my values for $A, B$ ...
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69 views

Why doesn’t the equation break when we put a value $x$ while solving partial fractions?

Let’s suppose we want to decompose $\frac {9}{9-x^2}$. We proceed and we get to solve $(3-x)a + (3+x)b = 9$ where I assumed $a$ and $b$ to be the numerators of the fractions. Then we assume with $x=3$ ...
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1answer
32 views

Integration by partial fractions problem

I'm having a lot of trouble with this integral. I don't know what to set A+B equal to. There's an x^4 in the numerator and I'm trying to figure out how to account for it. What am I doing wrong? Thanks....
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1k views

How can I calculate $\int\frac{x-2}{-x^2+2x-5}dx$?

I'm completely stuck on solving this indefinite integral: $$\int\frac{x-2}{-x^2+2x-5}dx$$ By completing the square in the denominator and separating the original into two integrals, I get: $$-\int\...
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1answer
40 views

Working partial fractions to solve for inverse Fourier Transfrom

I have a system where $H(w) = \dfrac{1}{(1-\frac{1}{4}e^{-jw})(1-\frac{1}{3}e^{-jw})}$. I need its inverse discrete Fourier transform. My thinking is that I could use partial fraction decomposition ...
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0answers
55 views

Closed form for infinite series $\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)\ldots(n+m)}$ [duplicate]

Let $m$ be a positive integer. Is there a closed form for the infinite series $\displaystyle\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)\ldots(n+m)}$? In the cases $m=1, 2$, one can use a partial fraction ...
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0answers
56 views

Mistake in the computation via partial fractions

This is a computation made in Titchmash's Introduction to Zeta functions. I was trying to reverse the computation. However, I kept missing factors. Consider $\frac{1- xyz^2}{(1-z)(1-xz)(1-yz)(1-xyz)}...
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2answers
29 views

How to separate an equation into partial fractions?

I am looking at a math question that has simplified this: into this: Can somebody explain the process for how this simplification was made? i.e. how does the denominator get broken down to those ...
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0answers
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About partial fraction decomposition in Michael Spivak's “Calculus”. Why degree 8?

I am reading Michael Spivak's "Calculus" now. I cannot understand why the degree of the polynomial in the picture below is 8. I think the degree is 7.
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About a statement of partial fraction in an answer

I'm reading this answer of The logic behind partial fraction decomposition, I think my question is too basic and not directly related to the answer so I don't comment there. I don't understand why: ...
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3answers
79 views

Partial fraction failure for $\frac{1}{\sin(x-a)\sin(x-b)}$

Here in this answer I have tried to perform partial fraction for $\frac{1}{\sin(x-a)\sin(x-b)}$ as follows: $$\frac{1}{\sin(x-a)\sin(x-b)}=\frac{p}{\sin(x-a)}+\frac{q}{\sin(x-b)} \quad(1)$$ ...
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1answer
30 views

Can this Fourier transformed function be transformed into partial fraction?

Hi I'm self learning stochastic process, I've come across a problem and found $$H=\frac{2+j(2w)}{(1-w^2)+j(2w)} \\ |H(iw)|^2=\frac{2^2+(2w)^2}{(1-w^2)^2+(2w)^2} \\$$ In attempt to find the power ...
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3answers
75 views

Prove that $\sum^n_{k=0}\frac{(-1)^k}{k+x}\binom{n}{k}=\frac{n!}{x(x+1)\cdots(x+n)}$.

Given the following formula $$ \sum^n_{k=0}\frac{(-1)^k}{k+x}\binom{n}{k}\,. $$ How can I show that this is equal to $$ \frac{n!}{x(x+1)\cdots(x+n)}\,? $$
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3answers
47 views

Partial fraction for complex roots using the second order polynomial

$\frac{(s^2 +s +1)}{(s^2+4s+3)(s+1)}$ the answer has to be in a $\frac{A}{(s+1)} + \frac{Bs+C}{(s^2+4s+3)}$ form. However i tried to solve it this way but end up with that there is no solution for ...
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1answer
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$2n$ th partial derivative of $\frac{1}{y(1+x^2)-1}$ with respect to $x$.

I need to find the $2n$ th derivative with respect to $x$ of the function $f = \frac{1}{y(1+x^2)-1}$. I tried differentiating util a pattern was founded, but that didn't happen. I think the $x^2$ is ...
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2answers
36 views

Partial Fraction: Already irreducible?

I have this partial fraction: $${3x+7}\over{(x-4)^2+25}$$ As far as I can tell, I do not think this can be decomposed. Is that a correct assumption? Sorry for the very short question, there isn't ...
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0answers
87 views

Integrating $\frac{x}{\sin x}$ using infinite products and fraction decomposition

I found this way of integrating $\frac{x}{\sin x}$ using infinite products and fraction decomposition. $$I=\int\frac{xdx}{\sin x}=\int\frac{xdx}{x\prod_{n\geq1}(1-\frac{x^2}{\pi^2n^2})}\\I=\int\prod_{...
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3answers
44 views

Can the quadratic formula be used when factorizing a denominator?

I’m doing partial fractions and need to factorize the denominator. They are quadratic. However there are some that aren’t so easy to factorize and my first choice was to use the quadratic equation to ...
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2answers
58 views

How can partial fractions be used for deductions?

Find partial fractions of the expression,$\frac{(x-p)(x-q)(x-r)(x-s)}{(x-a)(x-b)(x-c)(x-d)}$ . Hence deduce that; $\frac{(a-p)(a-q)(a-r)(a-s)}{(a-b)(a-c)(a-d)}+\frac{(b-p)(b-q)(b-r)(b-s)}{(b-a)(b-c)(...
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2answers
58 views

Partial fractions in calculus

Why are multiplicity expanded in the manner they are expanded? For example, $$ \frac{x^2}{(x-2)^2 \cdot (x-9)}=\frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{(x-9)}$$ Why is the $(x-2)^2$ expanded ...
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2answers
78 views

Partial Fraction of $\int \frac{ \left( \cos x + \sin 2x \right) \ \mathrm{d}x}{( 2 - \cos^2 x)(\sin x)}$

If $$\int\frac{ \left( \cos x + \sin 2x \right) \ \mathrm{d}x}{( 2 - \cos^2 x)(\sin x)} = \int \frac{A \ \mathrm{d}x}{(\sin x)} + B \int\frac{\sin x \ \mathrm{d}x}{ 1 + \sin^2 x} + C \int \frac{\...
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0answers
176 views

write as a single fraction in its simplest form.

I've been given this problem to solve and a little confused on how to go about it. Any help would be greatly appreciated. Here's the problem I need to solve: $$55+\frac{x^5}{(x-6)(x+1)}-\frac{7x}{x+...
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1answer
36 views

Mistake in partial fraction

Can anyone spot my mistake? $$\frac{1-\frac{1}{2}z^{-1}}{1+\frac{3}{4}z^{-1}+\frac{1}{8}z^{-2}}$$ Set $x = z^{-1}$ $$\frac{1-\frac{1}{2}x}{1+\frac{3}{4}x+\frac{1}{8}x^{2}}$$ Multiply by 8/8 and ...
4
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3answers
117 views

Is there any mistake in my approach for solving $ \int_0^{\pi/2} \frac{ \cos x}{3 \cos x + \sin x} \, dx $ ??

I had to evaluate this integral . $$ \int_0^{\pi/2} \frac{\cos x}{3 \cos x + \sin x} \, dx $$ Here is how I proceeded Dividing $N^r$ And $D^r$ by $\cos^3 x$ $$ \int_0^{\pi/2} \frac{ \sec^2 x}{3 \...