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

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Still getting wrong answer after trying to solve $x''(t)+4x(t)=t^2$ where $x(0)=1$ and $x'(0)=2$

I am trying to solve this differential equation: $$x''(t)+4x(t)=t^2,x(0)=1,x'(0)=2$$ The answer should be: $$x(t)=\frac{1}{4}t^2-\frac{1}{8}+\frac{9}{8}\cos{2t}+\sin{2t}$$ Which is also verified ...
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Partial fractions problem help

I need help with the following: $\frac{1}{(x-1)^2(x+1)}$ Using the "cover-up" method, I can identify the numerator of the fraction with the denominator $(x+1)$ fairly quickly, and it is $1/4$. Then ...
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Why I'm getting wrong answer? What is wrong in my solution?

I'm learning single variable calculus right now. Right now trying to understand integration with partial fraction. I'm confused in a problem from sometime. I think I'm doing right but answer in my ...
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4answers
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Find $\int\frac{x-1}{x^2-5x+6}dx$. Why my solution is different from book?

I'm learning single variable calculus right now. Right now trying to understand integration with partial fraction. I'm confused in a problem from sometime. I think I'm doing right but answer in my ...
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5answers
105 views

Is $(x^2 + 1) / (x^2-5x+6)$ divisible?

I'm learning single variable calculus right now and at current about integration with partial fraction. I'm stuck in a problem from few hours given in my book. The question is to integrate $$\frac{x^2 ...
2
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4answers
181 views

Calculus Integral from Partial Fractions

When you have an irreducible quadratic factor repeated you can get integrals that look like $\int \dfrac{dx}{(x^2+a)^m}$, where $m>1$, integer, and $a>0$. What is the best way to integrate this ...
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69 views

Method of partial fractions when denumerator cannot be factorized?

Suppose I'm given an expression: \begin{align*} P(x) = \frac{1+x}{1-2x-x^2}. \end{align*} The denumerator cannot be readily factorized, so I found the zeros, which are \begin{align*} \lambda_1 = -1 - ...
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Partial Fraction Decomposition: $\int_{0}^\infty \frac{xdx}{(x+\theta)^4} $

This is a simple calculus question, but it's been a while since I've had to use partial fraction decomposition. I need to integrate the following: $$\int_{0}^\infty \frac{xdx}{(x+\theta)^4} $$ I ...
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18 views

Partial fraction decomposition with exponential in denominator

Can the fraction below be decomposed ? $s$ is the Laplace variable and $T$ is a constant. $$\frac 1 {s (1-e^{-s T})}=?$$
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67 views

Help with Calculate integral

Find $\int^a_0 \dfrac{3x^2-ax}{(x-2a)(x^2+a^2)} dx$ I tried using partial fractions and the substitution $u=a-x$ but I haven't made any real progress. Please help.
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Integration by Partial Fractions $\int\frac{1}{(x+1)^3(x+2)}dx$

I'm trying to do a problem regarding partial fractions and I'm not sure if I have gone about this right as my answer here doesn't compare to the answer provided by wolfram alpha. Is it that I can't ...
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2answers
185 views

unresolved partial fraction decomposition

I'm having some trouble doing this partial fraction decomposition: $$\frac{1}{t^3-2t+1}$$ using Ruffini rule i get: $$\frac{1}{t^3-2t+1}= \frac{1}{(t-1)(t^2+t-1)}$$ i would like to decompose the ...
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40 views

partial fraction derivative question

So I have this partial fraction derivative question. I know how to solve it, but for some reason I keep swapping two numbers. Here is the problem: $$\int\frac{3-4x}{x^2+x}= ...
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49 views

Confusion about the integral $\int dT/(1-T^2)$

From some reference on the internet we have the following real valued function and its derivative: $$ M(T) = \frac{\sqrt{1-T^2}}{1+T} \quad \Longrightarrow \quad \frac{dM}{dT} = - M/(1-T^2) $$ The ...
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207 views

Contradiction between integration by partial fractions and substitution

Integration by substitution: $$\int \frac {dx}{x^2-1}$$ Let $x=\sec\theta$ and $dx=\sec\theta\tan\theta \,d\theta$ $$\int \frac {dx}{x^2-1} = \int \frac{\sec\theta\tan\theta ...
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1answer
34 views

Partial fractions, proper fraction with a perfect square factor

I think I have figured why the following is the case, I need however a confirmation that it is indeed so. Consider: $\displaystyle{\frac{x-1}{(x+1)(x-2)^2} \equiv \frac{A}{x+1} + ...
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1answer
62 views

Integration by partial fractions, reducing denominator through substitution

In the following case, suppose $b$ is a number and $b≠0$. Find $$ \int{1\over(x^2+b^2)^n}\mathrm{d}x$$ The textbook uses the following substitution: $$ x=bz$$ $$ \mathrm{d}x=b\mathrm{d}z$$ And ...
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1answer
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find the integral using Integration by partial Fractions

Here is my work for this problem...just wanted a check over and see if i missed anything Original Problem: $\int$ $\frac6{x^3-3x^2}$ F 6/x^3-3x^2= F 6/x^2(x-3) 6/x^2(x-3)= Ax+B/x^2+C/(x-3) ...
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2answers
64 views

$\int \frac{\left(2x^3-4x^2-x-3\right)}{\left(x^2-2x-3\right)}dx$ confusion

$$\int \frac{\left(2x^3-4x^2-x-3\right)}{\left(x^2-2x-3\right)}dx$$ =$$x^2+3\ln \left(x-3\right)+2\ln \left(x+1\right)+C$$ Where did the $x^2$ come from? What I did: Partial fraction then ...
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39 views

Complex partial fraction

Good day! I'm currently having an exercise on partial fractions... I know the basic of different methods in separating a fraction into smaller parts but I got confused when I encountered this ...
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28 views

Simple partial fractions questions with fractions in the numerator.

$$ \dfrac{1}{\left(x+1\right)\left(2x+1\right)}$$ Using the cover up rule, I received the answer of $$\dfrac{\dfrac{1}{2}}{2x+1}-\frac{1}{x+1}$$ or $$\dfrac{1}{2(2x+1)}-\frac{1}{x+1}$$ Answer ...
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34 views

Reciprocals of theta functions

I've spent the last few months with partial fraction expansions, and thought to create a function with simple poles over a lattice of zeros, like that of any of the Jacobi theta functions... but I ...
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3answers
42 views

How to Decompose into Partial Fractions?

I need to decompose (1/((s - 2)^(2) + 1^(2))) into partial fractions, but I am not sure how exactly. Here are my attempts: Attempt 1 (1/((s - 2)^(2) + 1^(2))) = (A/(s - 2)) + (B/(s - 2)^(2)) + ...
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5answers
128 views

Partial fraction of $\frac 1{x^6+1}$

Can someone please help me find the partial fraction of $$1\over{x^6+1}$$ ? I know the general method of how to find the partial fraction of functions but this seems a special case to me..
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80 views

Integral of $\int \frac{x+1}{(x^2+x+1)^2}dx$

I'm currently learning Calculus II and I have the following integral: Integal of $\large{\int \frac{x+1}{(x^2+x+1)^2}dx}$ I've tried with partial fractions but it led nowhere, I've tried with ...
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2answers
64 views

The partial fraction decomposition of $\dfrac{x}{x^3-1}$

I was trying to decompose $\dfrac{x}{x^3-1}$ into Partial Fractions. I tried the following: $$\dfrac{x}{(x-1)(x^2+x+1)}=\dfrac{A}{(x-1)}+\dfrac{B}{(x^2+x+1)}$$ $$\Longrightarrow A(x^2+x+1)+B(x-1)=x$$ ...
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58 views

How do I find $\frac {x^3}{x(x-3)}$ partial fractions?

How do I find $\frac {x^3}{x(x-3)}$ partial fractions? And in general, when the degree of the numerator is higher than the denominator's? Thanks in advance for your assistance!
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Partial fraction decomposition of $\frac{x-1}{x^3+x^2}$

Why the partial fraction expansion of $\frac{x-1}{x^3+x^2}$ is $$\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}$$ if I only have that $x^3 + x^2$ is the same as $x^2(x + 1)$?
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Partial fraction decomposition of $ \frac{\pi}{\sin \pi z}$

i want to prove the partial fraction decomposition: \begin{align}\frac{\pi}{\sin \pi z} = \frac{1}{z} + 2z\sum\limits _{n=1} ^{\infty} \frac{(-1)^n}{z^2-n^2} \end{align} wit the help of the partial ...
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1answer
20 views

Partial Fraction, numerator with a higher degree

I've been posed with the following question: $\int^3_1(x^2\div (2x+1)) dx$ I was able to determine via long division that: $x^2\div (2x+1)$ = $\frac12x-(\frac12x\div(2x+1))$ however, I can't ...
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1answer
27 views

Partial Fraction for rigorous understanding [ref. request]

I know usual techniques in partial fractions such as the ones described in Wiki. But I don't understand how this process works in a precise manner. I've seen in some complex analysis textbooks, ...
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63 views

Inverse Laplace Transform of $\dfrac{6s -19}{s^2 - 6s + 13}$.

I am trying to figure out the inverse laplace transform of $\dfrac{6s -19}{s^2 - 6s + 13}$. Looking at my table of Laplace Transforms in my textbook, it seems that either I must break up this fraction ...
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Integration of fractions and cancellation

How come $$\int\frac{3}{6+3x}\mathrm{d}x=\mathrm{ln}(2+x)+C$$ and not $\ \mathrm{ln}(6+3x)+C$ ? I understand you can cancel the fraction but why should this make a difference? I've tried searching ...
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1answer
36 views

Finding the partial fraction decomposition of $\frac{4s^2 - 5s + 2}{s^2(s^2 +9)}$

I am trying to find the partial fraction decomposition of $\dfrac{4s^2 - 5s + 2}{s^2(s^2 +9)}$ into something of the form $A\dfrac{1}{s} + B\dfrac{1}{s^2} + C\dfrac{1}{s^2+9} + D\dfrac{s}{s^2 + 9}$. ...
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53 views

Is this a valid partial fraction decomposition?

Write $\dfrac{4x+1}{x^2 - x - 2}$ using partial fractions. $$ \frac{4x+1}{x^2 - x - 2} = \frac{4x+1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x-2} = \frac{A(x-2)+B(x+1)}{(x+1)(x-2)}$$ ...
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1answer
55 views

Partial Fraction Using the Long Method

I've been trying to express $$\frac{x^2-13}{x^3-7x+6}$$ as a partial fraction, and, so far I have arrived at $$x^2-13=A(x-2)(x+3)+B(x-1)(x+3)+C(x-1)(x-2)$$. From this onwards, I substituted different ...
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1answer
21 views

Possible division by zero when repeatedly factoring and canceling (x-r) and evaluating the resulting polynomial at x=r…

The following proof is from 'Two Proofs of the Existence and Uniqueness of the Partial Fraction Decomposition', page 4. Lemma 2.1. Let $h(x), g(x) \in R[x]$ (real polynomials), $k$ a positive ...
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How can I integrate $\int \frac{dx}{(1+x^2)(2+x)}$ using partial fraction?

I did this: $$ \frac{1}{(1+x^2)(2+x)} = \frac{A}{1+x^2} + \frac{B}{2+x} \Rightarrow 2A+B+Ax+ Bx^2 = 1 $$ but I do not know what to do next. Every help will be appreciated.
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how to resolve this integral using partial fractions

I cannot solve this, please help me and explain me! $$\int \frac {2x-3} {x(2x+1)(2x-1)}\text{d}x$$ EDIT: I get this far $$ \frac {2x-3} {x(2x+1)(2x-1)} = \frac{A}{x} + ...
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308 views

Decomposition into partial fractions to compute an integral

I'm having problems with: $$\int_{-\infty}^{\infty}\frac{x^4+1}{x^6+1}dx$$ I was thinking: $\frac{x^4+1}{x^6+1}$ is an even function and the interval $(-\infty,\infty)$ is symmetric about 0, we ...
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39 views

partial fraction decomposition braindead

decompose $\frac{x^2-2x+3}{(x-1)^2(x^2+4)}$ the way my teacher wants us to solve is by substitution values for x, I set it up like this: (after setting the variables to the common denominator and ...
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1answer
27 views

Inversion of $z$-transform using partial fraction decomposition

I want to inverse a $z$-transform of this general form $$X(z) = \frac{b_0 + b_1z^{-1}+\cdots+b_Mz^{-M}}{a_0 + a_1z^{-1}+\cdots+a_Nz^{-N}}$$ where $M$ < $N$. In order to do this, I use partial ...
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Question in regard to solving for inverse laplace transform

I am having some confusion when it comes to solving for the inverse laplace transform. ( We are allowed the tables with the common values by the way). Il give an example. Take, ...
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422 views

How to solve certain types of integrals

I'm asking for a walk through of integrals in the form: $$\int \frac{a(x)}{b(x)}\,dx$$ where both $a(x)$ and $b(x)$ are polynomials in their lowest terms. For instance $$\int ...
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18 views

Getting Partial Fraction Decomposition Wrong

So I have done this solution: But it's wrong according to technology. I fail to see any error I've made in this solution, do you guys have any idea what's wrong?
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26 views

Show that the following expansion is valid for first $n$ terms

The question is as follows, I've gone ahead and attempted as much as I could of it twice, however the final answer that I'm receiving does not match. I'm omitting the partial fractions steps; if ...
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1answer
26 views

Simplification imaginary fractions

In an exercise, a partial fraction expansion has to be done. I have no problem with that, but one of the last steps includes a simplification as follows: \begin{equation*} \left( -\frac 12 - \frac 16 ...
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76 views

Evaluate the integral $\int \frac{dx}{x^3 + 2x^2 + 2x}$ of a rational function

Evaluate $$\int \frac{dx}{x^3 + 2x^2 + 2x}.$$ I have no idea how to approach this. I know how to solve rational functions with numerator as highest degree polynomial using division and remainder. ...
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1answer
31 views

Values tried for partial fraction decomposition

I'll explain my question with the following example from wikipedia. Suppose, we have a function: $$ f(x)=\frac{1}{x^2+2x-3} $$ Here, the denominator splits into two distinct linear factors: $$ ...
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1answer
28 views

Partial fraction question with squared

How could one apply partial fraction decomposition to $$\left(\frac{s}{s^2+4}\right)^2$$ I tried to separate by doing $$\frac{A}{s^2+4} + \frac{B}{s^2+4}$$ and I got strange solution $A+B=0$ and ...