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

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Integrate algebraic fraction with constant on top?

I understand that if you have $\int \frac{1}{x + 1} dx$ you simply do $\ln(x + 1) + C$. Now I'm slight confused because in my text book, $\int \frac{31}{x - 4} dx$ evaluates to $31\ln(x - 4)$ but ...
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4answers
258 views

Easy partial fraction decomposition with complex numbers

There is an easy method to perform a partial fraction decomposition - described here, under the "Repeated Real Roots" title, for the coefficient A2. The problem is ...
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3answers
50 views

How to compute $\int \frac{1}{(x^2+1)^2}dx$?

Suppose we know $\int \frac{1-x^2}{(x^2+1)^2}=\frac{x}{x^2+1}+C$ How to compute $\int \frac{1}{(x^2+1)^2}dx$? I tried writing it as ...
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3answers
58 views

How do you use partial fraction decomposition to break up $1/(s+4)^2$?

How do you use partial fraction decomposition to break up $1/(s+4)^2$? The usual method isn't giving me an answer.
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1answer
19 views

Question about partial fractions and the order of two linear factors

I have a question on the topic of partial fraction and decomposition. Lets say I have the integral of $\frac{1}{(x^2 - x - 2)}$. When I want to find out the A and B values, I first have to get linear ...
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1answer
24 views

Use Laplace Transformations to solve $y''+2y'+5y=3e^{-x}sin(x)$, with $y(0)=0$, $y'(0)=3$

I've gotten this far and I cannot proceed: $L[y]=\frac{L[3e^{-x}sin(x)]+3}{p^2+2p+5}= \frac{3}{((p+1)^2+1)(p^2+2p+5)}+\frac{3}{p^2+2p+5}$ I'm finding it impossible to find the inverse to solve for ...
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2answers
58 views

Sum of series (telescoping)

I have the following problem: $$\sum^{\infty}_{n=1}\dfrac{2}{\left(n+2\right)\sqrt{n}+n\sqrt{n+2}}$$ I should find the sum of this sequence. I tried to simplify but it does not work.
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1answer
48 views

Show $\frac{\pi}{\cos \pi z}=4\sum\limits_{n=0}^\infty \frac{(-1)^n(2n+1)}{(2n+1)^2-4z^2}$ using partial fraction decomposition of $\cot$ and $\tan$

I am asked to show that $\frac{\pi}{\cos \pi z}=4\sum\limits_{n=0}^\infty \frac{(-1)^n(2n+1)}{(2n+1)^2-4z^2}$. Here is what I have got so far, $\frac{\pi}{\cos (\pi ...
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1answer
45 views

(Laplace Method) $y'' - 4y' = 6e^{3t} - 3e^{-t}$

For this problem $y(0) = 1$ and $y'(0) = -1$ I need to solve this problem using this: \begin{align*} y(t) &\longrightarrow Y(s)\\ y'(t) &\longrightarrow sY(s) - y(0)\\ y''(t) ...
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2answers
22 views

Which rules are used to make function like one in Laplace Transformations table?

I have function like this: $$\frac{s^2+3s+3}{(2s^2+7s+7)} $$ It needs to be brought to the level of Laplace Transformations from table, like these two: $$\frac{a}{(s-b)^2 + a^2} $$ ...
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3answers
40 views

How to come from $\frac{z^{2}}{z^{2}+1}$ to $\frac{z/2}{z+i}+\frac{z/2}{z-i}$

As title says, how to come from $\frac{z^{2}}{z^{2}+1}$ to $\frac{z/2}{z+i}+\frac{z/2}{z-i}$? Here is what I did: $\frac{z^{2}}{z^{2}+1}=1-\frac{1}{z^{2}+1}=1-\frac{1}{(z-i)(z+i)}$
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1answer
49 views

Partial fraction of a generating function

I am solving a recurrence relation $a_0 = a_1 = a_2 = 1, a_{n+3} = a_{n+2} − 2a_{n+1} − 4a_n$ for $n \ge 0$ I got a generating function for this sequence $f(x) = \frac{2x^2+1}{4x^3+2x^2-x+1} $ ...
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0answers
20 views

How do we know the form of a partial fraction expansion?

If you take a quick gander at the table on this page: http://tutorial.math.lamar.edu/Classes/CalcII/PartialFractions.aspx You'll notice there are some rules about what the form of a partial fraction ...
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1answer
25 views

Can I plug imaginary unit to solve partial fraction?

for example, I have partial fraction to solve: $$ 1/(x^2+1)(x^2+4)=A/(x^2+1)+B/(x^2+4)\ $$ then I need to solve $$ 1=A(x^2+4)+B(x^2+1) $$ Can I plug x=i and 2i so that I can get the value of A and ...
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1answer
24 views

Use partial fractions to write $\frac{1}{x^2 + x – 1}$ as $\frac{A_1}{x – α_1} + \frac{A_2}{x – α_2}$ and write it as a power series

Find the roots $α_1$, $α_2$ of $x^2 + x – 1$ and use partial fractions to write $\frac{1}{x^2 + x – 1}$ as $\frac{A_1}{x – α_1} + \frac{A_2}{x – α_2}$ , for suitable $A_1, A_2$. Using the power series ...
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1answer
24 views

Why Bx+C for irreducible quadratic but not perfect square? (Partial fractions)

When you partial fraction something like $$\frac{x+2}{(x+3)(x+2)^2}$$ you make it $$\frac{a}{(x+3)}+\frac{b}{x+2} + \frac{c}{(x+2)^2}$$ but when you have $$\frac{10x^2+12x+20}{(x-2)(x^2+2x+4)}$$ you ...
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4answers
50 views

Integration Partial Fractions

I have the following problem: $$\int\left(\frac{x+2}{x^2+x+1}\right)dx$$ I received this by simplification of another integral. But my question is how to procede from here. Is there a way to simplify ...
4
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3answers
99 views

Integral of $\frac{1}{x\sqrt{x-1}}$ using partial fractions.

Find $$\int\frac{1}{x\sqrt{x-1}}\ dx.$$ I have attempted to use the $A/x + B/\sqrt{x-1}$ method. That does not work. I have tried a substitution with $t= \sqrt{x-1}$ and $t^2 +1 = x$. Finding the ...
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1answer
50 views

$\int_0^5 \frac{dx}{x^2-x-2}$

I am having some difficulty with this problem. I am getting a finite answer but when I put the equation into wolfram alpha to check my answer it says that the integral does not converge.Here is what I ...
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4answers
59 views

How was this factoring of $1-2x-x^2$ achieved?

If I factor $1-2x-x^2$ using the quadratic formula I get $$x=\frac{2\pm \sqrt{4-4(-1)(1)}}{2(-1)}$$ $$x=\frac{2\pm \sqrt{8}}{-2}$$ $$x=-1 \pm \sqrt{2}$$ Let $\alpha = -1 +\sqrt{2}$ and ...
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4answers
82 views

Partial fraction decomposition for $\int\frac{3x^3+1}{x^3-x^2+x-1}dx$

I have been at this for hours and I don't know what I am doing wrong. It's partial fraction decomposition that I am doing but I just can't seem to get what I am supposed to ...
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2answers
27 views

Partial fraction decomposition of $\frac{21}{s^{2}+4}$ for inverse-Laplace transform

So I have this number which I want to do inverse-Laplace transformation on, which is kind of complicated. So it would be easier to do some partial fraction decomposition first. I am trying to do the ...
0
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2answers
50 views

Solving for $x$ in a Laplace equation

So I have this Laplace equation: $$s^{2}x+4sx+5=\frac{s}{s-1}$$ And I want to solve for $x$. My result is the following: $$x = \frac{5-4s}{s^{3}+3s^{2}-4s}$$ Which is also the same answer that for ...
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1answer
38 views

Find partial fractions of $\frac{x^2-1}{x^4+1}$

I tried to figure out how WA found the partial fractions of $\frac{x^2-1}{x^4+1}$: Help is Appreciated.
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2answers
39 views

Comparing Coefficients for Partial Fractions

See this simple example : $$\frac{x+1}{(x-1)(x-2)}\equiv \frac{A}{(x-1)}+\frac{B}{(x-2)}$$ Then we can get $x+1 \equiv A(x-2)+B(x-1)$ for $x \neq 1 ,2$ My Question : Is it correct to put $x=1$ ...
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2answers
41 views

Partial Fraction confusion

How to solve this using partial fraction $$\frac{x^2+7}{(2x-1)(x-1)}$$ I am using $\frac{A}{2x-1}+\frac{B}{x-1}$ but not getting it
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2answers
69 views

How to get the partial factor representation of $\frac{1}{x^2+4x+1}$? [closed]

I've been trying to factor this, but I don't see it. What is the partial fractions representation of $$\frac{1}{x^2+4x+1}$$?
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1answer
30 views

Solving an IVP with Laplace Transforms

I'm trying to solve the following IVP (differential equations) with the Laplace Transform method: \begin{cases} y''+9y=36t\sin(3t)\\ y(0) = 0\\ y'(0) = 3 \end{cases} After taking the ...
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1answer
45 views

Integration of rational functions by partial fractions

I have the following integral : $$\int \frac{x^2+2x+3}{x^2-5x+6}~dx$$ Although I tried to solve it by partial fractions, I could not come up with an appropriate answer. My question is more about the ...
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4answers
95 views

Finding the infinite Sum of a series: $\sum\frac1{n(n+1)(n+2)}$

Find the infinite Sum of the series with general term $\frac{1}{n(n+1)(n+2)}$. I decomposed the fraction upto this $1/(2n)-1/(n+1)+1/(2n+4)$. But I find no link about cancelling terms. So how should ...
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2answers
43 views

Inverse Fourier Transform

I've got a problem where I need to find the IFT of $$F(\omega) = \frac{1 + i\omega}{6-\omega^2+5i\omega}$$ I've been trying to solve it through partial fractions, but that gives us $$\frac{1 + ...
4
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3answers
162 views

Integral using partial fractions

I am having trouble using partial fractions to evaluate $$\int \frac{6x}{x^3-8} dx.$$ I can find the denominator but using equations to find the numerator is difficult.
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2answers
56 views

Tedious fraction decomposition integral

I'm having this integral to resolve using fraction decomposition: $$\int\frac{1}{(x^2-1)^2} \, dx$$ This gives me the following: $$\frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x-1} + ...
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0answers
19 views

Gamma partial fractions

how convert this fraction to partial fractions .... (to take Laplace-Stieltjes inverce)... (gamma function) $${(λ_2^α (λ_1^α-(λ_1+s)^α)\over(λ_1+s)^α (λ_2+s)^α} ={A=?\over(λ_1+s)^α ...
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1answer
49 views

3rd degree polynomial fraction decomposition

Was solving some differential equations and came upon this integral: $$\int\frac 1{x(x+1)^2} dx$$ Looked it up on wolframalpha and it can be decomposed to: $$\frac ...
2
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7answers
111 views

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
42 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
38 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
36 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
40 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
43 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
34 views

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
33 views

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
43 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
68 views

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
votes
3answers
48 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: ...
1
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4answers
102 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 $\frac{Cx+d}{x^2+1}$ ...
3
votes
5answers
97 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 ...
3
votes
3answers
64 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 ...