Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [partial-fractions]

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

Rewriting rational function in the form of partial fractions is often useful when calculating integrals. The possibility of decomposing a rational function into a sum of simplified fractions is guaranteed by the fundamental theorem of algebra.

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Can all real polynomials be factored into quadratic and linear factors?

So I understand how to do integration on rational functions with a linear and a quadratic denominator, and I understand how to do a partial fraction decomposition, but I was wondering what happens if the polynomial is higher degree and can't be…
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Integration by partial fractions; how and why does it work?

Could someone take me through the steps of decomposing $$\frac{2x^2+11x}{x^2+11x+30}$$ into partial fractions? More generally, how does one use partial fractions to compute integrals $$\int\frac{P(x)}{Q(x)}\,dx$$ of rational functions ($P(x)$ and…
Finzz
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How does partial fraction decomposition avoid division by zero?

This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example: $$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$ Multiplying both sides by $x(x+1)$ we…
Argon
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Computing $\int_0^\infty\frac1{(x+1)(x+2)\cdots(x+n)}\mathrm dx $

I would like to compute: $$\int_0^\infty\frac1{(x+1)(x+2)\cdots(x+n)}\mathrm dx $$ $$ n\geq 2$$ So my question is how can I find the partial fraction expansion of $$\frac1{(x+1)(x+2)\cdots(x+n)}\ ?$$
Chon
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The logic behind partial fraction decomposition

In the general case of any function would be interesting but my question is concerning the general case of polynomials with integer powers. I can use the method of partial fractions in the simple case required for an introductory course on…
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Derivation of the general forms of partial fractions

I'm learning about partial fractions, and I've been told of 3 types or "forms" that they can take (1) If the denominator of the fraction has linear factors: $${5 \over {(x - 2)(x + 3)}} \equiv {A \over {x - 2}} + {B \over {x + 3}}$$ (2) If the…
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If $A$ and $B$ are positive constants, show that $\frac{A}{x-1} + \frac{B}{x-2}$ has a solution on $(1,2)$

I have problem which I couldn't figure out how to solve; If $A$ and $B$ are positive constants, show that $$0=\frac{A}{x-1} + \frac{B}{x-2}$$ has a solution on the open interval $(1,2)$. If you support your answers with rigorous proof, I appreciate…
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When is the series $\sum_{n=1}^\infty \frac1{a n^2 + b n + c}$ rational?

Let $a,b,c$ be integers such that $a\neq 0$ and $$ a n^2 + b n + c \neq 0 $$ for all positive integers $n$. (a) Prove that if there exists a positive integer $k$ such that $$ b^2 - 4ac = k^2 a^2, $$ then $$ \sum_{n=1}^\infty \frac{1}{a n^2 + b n +…
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Combinatorial Proofs of Real Analysis Identity

In this question, a proof using real analysis is given of the following identity: $$ \sum_{n=1}^{\infty} \frac{(n-1)!}{n \prod_{i=1}^{n} (a+i)} = \sum_{k=1}^{\infty} \frac{1}{(a+k)^2}$$ Is there a combinatorial proof of this identity? Is so, does…
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Why does partial fraction decomposition always work?

Say you have a function $p(x)/q(x)$ for some polynomials $p(x)$ and $q(x)$ and $p$ has a lower degree than $q$. Say $q$ has degree three and $p$ has degree two. If you partially decompose it, you'll get three variables $A$,$B$,$C$ and three…
dfg
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Partial Fractions: Why does this shortcut method work?

Suppose I want to resolve $1/{(n(n+1))}$ into a sum of partial fractions. I solve this by letting $1/{(n(n+1))} = {a/n} + {b/(n+1)}$ and then solving for $a$ and $b$, which in this case gives $a=1$ and $b=-1$. But I learnt about a shortcut method.…
Ram Keswani
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Partial fraction expansion for non-rational functions

With regard to this answer to an inverse Laplace transform question, I derived the following result: $$\frac1{i 2 \pi} \int_{c-i \infty}^{c+i \infty} ds \, e^{s t} \Gamma(s)^2 = 2 K_0 \left ( 2 e^{-t/2} \right ) - t I_0 \left ( 2 e^{-t/2} \right )…
Ron Gordon
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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 final answer would be: $$\frac{-4}{x+1} -…
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What is the Mathematical Property that justifies equating coefficients while solving partial fractions?

The McGraw Hill PreCaculus Textbook gives several good examples of solving partial fractions, and they justify all but one step with established mathematical properties. In the 4th step of Example 1, when going from: $$1x + 13 =…
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The existence of partial fraction decompositions

I'm sure you are all familiar with partial fraction decomposition, but I seem to be having trouble understanding the way it works. If we have a fraction f(x)/[g(x)h(x)], it seems only logical that it can be "split up" into A/g(x) + B/h(x) for some A…
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