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

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Integrate $\int \frac{x\cos x}{\sin^2x}dx$

$$\int \frac{x\cos x}{\sin^2x}dx$$ $$\int \frac{x\cos x}{\sin^2x}dx=\int \frac{x\cos x}{1-\cos^2x}dx=\int \frac{x\cos x}{(1-\cos x)(1+\cos x)}dx$$ How can I find the two fractions? if there are ...
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0answers
37 views

Methods for solving for partial fraction coefficients.

When I present partial fraction decomposition, there are typically two approaches to solve for the coefficients of the fractions: Clear the fractions, multiply out the products, and equate ...
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1answer
35 views

Why do difference of squares partial fractions have to be decomposed this way?

Why do you have to factor out $-1$ here? $$\frac{2000}{(10-h)(10+h)}$$$$=\frac{A}{10-h}+\frac{B}{10+h}$$ Decomposing this finds A annd B to be 100, which is wrong. Symbolab and Wolfram Alpha factor ...
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3answers
27 views

Substitution and Partial Fractions (Integration)

$$\int\frac{dx}{x-\sqrt[4]{x}}$$ given the substitution $x=u^{4}, dx=4u^{3}du$ $$=\int\frac{4u^{3}du}{u^{4}-u}=\int\frac{4u^{3}du}{u(u^{3}-1)}=\int\frac{4u^{2}du}{(u^{3}-1)}$$ At this point I ...
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1answer
31 views

Decompose $\frac{x^4 + 5}{x^5 + 6x^3}$ (partial fraction decomposition)

Write out the form of the partial fraction decomposition of the function. Do not determine the numerical values of the coefficients. $$\frac{x^4 + 5}{x^5 + 6x^3}$$ So I factored the ...
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0answers
41 views

Partial Fractions Integration Help [closed]

So this is the integral and i have tried going through it many times and i can't seem to figure out where i went wrong. The answer in the box is the final answer i came up with. I've used up all my ...
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3answers
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How do you solve B and C for $\frac{s-1}{s+1} \frac{s}{s^2+1} = \frac{A}{s+1} + \frac{Bs+C}{s^2+1}$?

How do you solve B and C for $\frac{s-1}{s+1} \frac{s}{s^2+1} = \frac{A}{s+1} + \frac{Bs+C}{s^2+1}$ ? $A = \left.\frac{s^2-s}{s^2+1} \right\vert_{s=-1} = \frac{1-(-1)}{1+1}=1$
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2answers
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Repeated Root Partial Fraction Decomposition: Derivative Aproach

I am trying to solve for H1, I was able to get the Coefficients for B,C, and D. Yet, I have forgotten how to solve for A. All I can remember is that one must take the derivative of both sides. After ...
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2answers
39 views

Partial fraction in two variable problem

How to write partial fraction of $$\frac{12m-n-3mn+7}{5m-2n-2mn+5}$$ I just write first and second denominator: $5-2n$ and $m+1$.
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6answers
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Solving integral using trig substitution $\tan(x/2)=t$

I have problems with solving the following integral: $$ \int{{\sin x - \cos x}\over {\sin x + \cos x}} \, dx$$ Could anybody please help me to find the solution and show me the method how it can be ...
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3answers
32 views

Is it possible to solve following integral with partial-fraction decomposition?

I have a integral $$\int_0^1\frac{-x^2+4x+4}{x^2-4}~dx$$ Which I changed to $$\int_0^1\frac{-x^2+4x+4}{(x-2)(x+2)}~dx$$ But I don't know how to change numerator to have lesser polynom degree than the ...
3
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3answers
42 views

How to properly set up partial fractions for repeated denominator factors

I was just trying to solve a problem that had the following item which I needed to split into separate generating functions: $$\frac{x}{(1-2x)^2(1-5x)}$$ I had assumed I needed to split it into: ...
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4answers
95 views

Partial Fraction Decomp. Why repeated factors. And why the (Cx+D) for quadratics?

1) First, why are powers of linear factors repeated? For example, $(x+1)^2$ gets $\frac{A}{(x+1)}$ and $\frac{B}{(x+1)^2}$ 2) Why does a quad factor like $x^2+1$ get a term $\frac{Cx+d}{x^2+1}$ ...
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5answers
85 views

Integral $\int\left(\frac{1}{x^4+x^2+1}\right)dx$

Someone can halp me to solve this integral: $$\int\left(\frac{1}{x^4+x^2+1}\right)$$ solution$$\frac{1}{4}\ln\left(\frac{x^2+x+1}{x^2-x+1}\right)+\frac{1}{2\sqrt3}\arctan\frac {x^2-1}{x\sqrt3}$$ I ...
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3answers
63 views

How to evaluate the following integral, $\int\frac{x \, dx}{x^2+2x+17}$?

I am new to integration. This function is kinda tricky for me : $$\int\frac{x \, dx}{x^2+2x+17}$$ I came up with following three approaches: Partial fraction decomposition, but I can't factor the ...
3
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2answers
111 views

Evaluate the integral $\int \frac{dx}{x(x+1)(x+2) \cdot \cdot \cdot (x+n)}$

I found the integral in this pdf. $$\int \frac{dx}{x(x+1)(x+2) \cdot \cdot \cdot (x+n)}$$ My first thought was to use partial fraction decomposition. Namely, $$ \begin{align*}\prod_{k=0}^n ...
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1answer
29 views

A Partial fraction expansion questions about Laplace transform

I am learning signals and systems. Our teacher give us the following answer, it's about Laplace transform . But I can't figure out the second line, the calculation of k1,k2,k3,k4. why they can be ...
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1answer
44 views

partial fractions with numerator and denominator of the same degree

I tried using partial fractions but wasn't sure what to do with the ${x^2}$: $$\int \frac{x^2}{3x^2 - 4} dx$$
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2answers
36 views

Inverse Laplace problem using partition fraction

Hello I am solving inverse Laplace transform using partial fraction. The question is: $$ X(s) = \frac{10(s+1)}{s(s^2+4s+8)} => \frac{10(s+1)}{s((s+2)^2+4)} $$ $$ \frac {C1} {s} + ...
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2answers
112 views

How to evaluate an integral from Griffiths

I'm working through Griffith's Electrodynamics book during the winter break, and I'm having trouble on evaluating this integral from problem 2.7 of Introduction to Electrodynamics 4th edition. I have ...
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1answer
29 views

Laurent series in various regions

I have the question: "Find the Laurent series which represents the function $$ f(z) = (z^2 - 1)/(z + 2)(z + 3)\ $$ in the regions (i) $\mid z\mid < 2\ $ (ii) $ 2 < \mid z\mid < 3\ $ ...
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1answer
57 views

How can I calculate $\int \frac1{x \cos x}\,dx$ ? / An issue wit an ODE.

How can I calculate this integral? $$\int \frac1{x \cos x}\,dx$$ Actually I was lost in this differential equation $$y' = -\frac{(x+2) \sin y}{x \cos x}$$ so I'd be glad if you could help me ...
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1answer
21 views

A Partial fraction expansion questions.

I am learning signals and systems. I solve the problem and reach the equation (5.164). How can I work out the value of $A$ and $B$? I do the partial fraction expansion and yields
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0answers
37 views

Inverse Laplace Transform of $F(s) = \frac{(3s^2+9s+14)*e^{-5s}}{(s^3+4s^2+7s)}$

Find the inverse Laplace Transform of $F(s) = \frac{(3s^2+9s+14)*e^{-5s}}{(s^3+4s^2+7s)}$ I have found the Simplified $F(s) = (\frac2s+\frac{s+1}{(s+2)^2+3})*e^{-5s}$ I am having trouble figuring ...
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0answers
16 views

Decompose into Partial Fraction Series

Given two positive integers $m$ and $k$, complex $a$ and the rational polynomial $$ p(z) = \frac{1}{z^{m+k} + a z^{m} + a z^{k} + 1}. $$ Is there a partial fraction expansion over the complex numbers, ...
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1answer
37 views

What rule is used for this simplification?

$$ \frac{8}{(s+1)^2 + 2^2} \times \frac{1}{s} = \frac{8}{5} - \frac{1}{s} + \frac{16}{10}\times \frac{s+1}{(s+1)^2 + 2^2} + \frac{8}{10}\times \frac{2}{(s+1)^2 + 2^2} $$
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1answer
34 views

How to obtain this partial fraction decomposition?

I am studying Laplace transforms right now and got stuck at this step that involves a weird partial fraction decomposition. It looks like the instructor skipped a bunch of steps and assigned ...
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2answers
38 views

Help with improper integrals containing partial fractions

Specifically, $$\int_1^\infty \frac {24}{8x(x+1)^2} dx$$ and $$\int_3^\infty \frac {1}{t^2 - 2t} dt$$ Both of these problems are supposed to converge, but I keep getting infinity in my answer. For ...
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2answers
58 views

Evaluate the integral $\int \frac{dx}{2 \sin x - \cos x + 5}$

I'm trying to evaluate the following integral: $$ \int \frac{dx}{2 \sin x - \cos x + 5}.$$ This is in a set of exercises following a chapter on partial fractions, so I imagine there is a ...
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1answer
11 views

Partial fraction decomposition in $\mathbb C (X)$ of real fraction

Let $F={P\over Q} $ be a fraction with $deg(P)< deg(Q)$ and such that $P$ and $Q$ are polynomials with real coefficients. Suppose $Q$ has the form $Q=(X-a)(X-z)(X-\bar z)$ with $a\in \mathbb R$ ...
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1answer
60 views

Integrate $\int\frac{x+1}{(x^2+7x-3)^3}dx$

How can i solve something like that? $$\int\frac{x+1}{(x^2+7x-3)^3}dx$$ How should I start? Should I try rewrite it in partial fractions?
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2answers
54 views

Definite Integral involving reciprocals of logs

Integrate $$\int_2^{4e} \frac{1}{x \ln(x+1)}\,dx $$ I have tried partial fractions, u substitution and parts but i cant get the final answer out. my main problem is dealing with the $x$ and $x+1$ ...
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4answers
66 views

What is the integral of $\frac{x-1}{(x+3)(x^2+1)}$?

I've worked with partial fractions to get the integral in the form $$\int\frac{A}{x+3} + \frac{Bx + C}{x^2+1}\,dx$$ Is there a quicker way?
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2answers
25 views

partial fraction for complex roots

While solving Laplace transform using Partial fraction expansion. I have confusion in solving partial fraction for complex roots. I have this equation $$ \frac {2s^2+5s+12} {(s^2+2s+10)(s+2)}$$ ...
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2answers
24 views

Finding the inverse laplace of this function: $ F(s)= \frac{s+8}{s^{2}+4s+5}$

Im trying to find the inverse laplace of : $ F(s)= \frac{s+8}{s^{2}+4s+5}$ I reached the following: $$ F(s)= \frac{s}{(s+2)^{2}+1} + 8 \times \frac{1}{(s+2)^{2}+1}$$ Now i have the 2nd term in the ...
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0answers
25 views

Inverse Laplace Transform by Partial Fraction Expansion

I've been trying to solve this partial fraction for a Laplace transformation but I can't. Is there any way to solve it? $$\frac{(s-t)^2}{((s-t)^2-1)((s+1)^2+4)}$$ Could somebody help, I've been ...
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3answers
46 views

Express $\frac{9x}{(2x+1)^2(1-x)}$ as a sum of partial fractions with constant numerators. Answer doesn't match with solution provided in book.

Express $\frac{9x}{(2x+1)^2(1-x)}$ as a sum of partial fractions with constant numerators. Answer doesn't match with solution provided in book. My method: ...
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0answers
25 views

Partial Fraction Decomposition (Distinct factors in denominator that don't have X in them)

When performing a partial fraction decomposition on the expression: $$ \frac{11x^2 - 5x - 10}{5x^3 - 5x^2} $$ The denominator factored into: $$ x^2, 5, (x-1) $$ but I got the wrong answer by ...
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1answer
31 views

Confused on algebraic step to derive a rule for partial fractions

I am trying to understand the partial fraction identity $$\frac{5x+4}{(x^{2}+9)(x-1)} = \frac{A}{x-1} +\frac{Bx + C}{(x^{2}+9)}$$ I transformed it to be equal to $$\frac{A}{x-1} + \frac{B}{x+3i} + ...
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1answer
22 views

Question on uniqueness in the proof used for the partial fraction algorithm

Partial fraction algorithm Let $\mathbb{F}$ be a field, and let $f(x)$, $a(x)$, $b(x)$ be polynomials in $\mathbb{F}[x]$ such that $a(x)$ and $b(x)$ are coprime and $\deg f < \deg a + \deg b$. ...
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1answer
18 views

Find a common denominator

I'm trying to integrate a function but first I need to find a common denominator for: $\frac{A}{x-8}\ $ + $\frac{B}{x+1}\ $ + $\frac{C}{x-1}\ $
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2answers
30 views

Partial Fraction for $x^2/(x^2 +1)^2$

$$\begin{aligned} \frac{\ x^2}{(x^2+1)^2} \\ \ \end{aligned}$$ I am new to partial fractions and this is what I have so far: $$\begin{aligned} \dfrac{(x^2)}{(x+1)^2} = ...
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4answers
223 views

Write a formula as a sum of fractions with constant numerators

I'm supposed to write this formula: $$\frac {9a + 43}{a^2 + 9a + 20}$$ As a sum of fractions with constant numerators as: $$\frac {7}{a+5} + \frac {2}{a+4}$$ The first step is of course: $$\frac ...
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3answers
51 views

How is this is an identity?

So we have started studying partial fractions.The book teaches two methods: By equating coefficients By utilizing the fact that when a rational fraction is decomposed to partial fractions it is an ...
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0answers
23 views

Partial fraction decomposition question

How do I use partial fraction decomposition to split up the following: $$\frac{1}{(1-q_1z)(1-q_2z)}$$ where both $q_1$ and $q_2$ are probabilities, and $q_1 = 1 - p_1$ and $q_2 = 1 - p_2$ The ...
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1answer
40 views

Decompose into Partial Fraction. Image Added. [duplicate]

I really had no idea how to write these questions out without copying and pasting them onto here, so I am sorry for that..I hope adding a picture is fine. I would appreciate any kind of help, and if ...
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0answers
13 views

Division of a monomial by a polynomial

Consider the following division of a monomial by a polynomial $$ \frac{x^n}{\prod_{i=1}^m (x-b_i)^{c_i}}$$ where $n$ and the $c_i's$ are given integers and $n>\sum_i c_i$. It is clear that one can ...
2
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3answers
212 views

How can I evaluate $ \int \frac{dx}{x(1+x^3)(1+3x^3)}$? [closed]

I want to solve this problem but I have no clue on break the denominator: $$ \int \frac{dx}{x(1+x^3)(1+3x^3)}$$ I have tried breaking the denominator into partial fractions but failed to do so.
1
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1answer
35 views

Convert partial fraction to continued fraction?

Lets say you have a partial fraction of the form: $$ f(x) = a_0 + \sum_{n=0}^{\infty} \frac{a_n}{\lambda_n + x} $$ Can anyone explain to me, in mildly plain English, how to convert this partial ...
0
votes
2answers
33 views

Evaluate: $ \displaystyle\int \frac{9x^2 - 30x - 73}{(x - 4)(x^2 - x - 12)} \; dx$ Partial fractions

Evaluate: $\; \displaystyle\int \frac{9x^2 - 30x - 73}{(x - 4)(x^2 - x - 12)} \; dx$ I couldn't find A, how to find A in the partial fractions below? $$\int \frac{9x^2 - 30x - 73}{(x - ...