# Questions tagged [partial-fractions]

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

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### Partial fractions with a repeated factor [duplicate]

I am looking to find a derivation, or a intuitive explanation as for why a partial fraction with a repeated factor needs to include a factor in the expansion for each power possible. How does one ...
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### Determining denominator of partial fractions

Before, integrating, we can often split a fraction into its partial fractions to make the integration process significantly more simple. However, I have realised that this fraction we can split can ...
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### Justification for equality in partial fraction expansion from generatingfunctionology by Herbert S. Wilf

The problem is from generatingfunctionology by Herbert Wilf on page 4. My question is not about the process of getting the generating function (they do a good job in this post) but rather where the ...
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### How to compute the partial fraction decomposition of $\frac{6}{x^4(x+1)}$?

How do I compute the partial fraction decomposition of $\frac{6}{x^4(x+1)}$? I let $$\frac{6}{x^4(x+1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4} + \frac{F}{x+1}$$ When I let $x=-1$...
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### How can i do the following partial decomposition?

I need to prove that: $$\frac{1}{(x-a)(x-b)} = \frac{1}{(b-a)(x-b)}- \frac{1}{(b-a)(x-a)},$$ and I must note that I need to go from the left expression to the right (because of the exercise). So, I ...
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### Finding the Inverse Laplace transform using partial fractions

Problem: Given: $$Y(s) = \dfrac{3s^2 + 6s+ 84} {( s+1 )(s-2)(s^2+ 2s+10) }$$ Find $y(t)$ by computing the inverse Laplace transform. Answer: To do this, we use the technique of partial fractions. \...
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### Partial Fractions in Integrations with complex Factorization.

I know how to integrate $\sqrt{tan}$, using the "classical approach" which I know has been illustrated on another question before. However, I have just recently learnt to factorise using ...
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### Show that $1+\sum_{n=1}^{\infty} \frac{1}{n^2 (n+1)} = \frac{\pi^2}{6}$

Show that $$1+\sum_{n=1}^{\infty} \frac{1}{n^2 (n+1)} = \frac{\pi^2}{6}$$ Proof By Partial Fraction Decomposition $$\frac{1}{n^{2} \left(n + 1\right)}=\frac{-1}{n}+\frac{1}{n^{2}}+\frac{1}{n + 1}$$ ...
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### Integration help for a beginner

This might be a silly question but I am a beginner in calculus so I really do not understand. When solving the problem $\int\frac{5+x}{\sqrt{16−(x+4)^2}}dx$, why we can’t divide it into the two ...
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### Extracting the sequence generated by $\frac{1}{(1-x)(1-x^2)(1-x^3)}$ [duplicate]

So I need the general formula for the sequence generated by the generating function $$\frac{1}{(1-x)(1-x^2)(1-x^3)}.$$ My idea was the decompose this into partial fractions and thus easily deduce ...
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### Computing $I_n=\int_0^1 \frac{x^n}{6+x-x^2} d x$

After reading the reduction formula in the post, I am curious about the closed form of the integral $$I_n=\int_0^1 \frac{x^n}{6+x-x^2} d x$$ I first resolve the integrand into two partial fractions ...
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### Integrate reciprocal of polynomial (coefficient of partial fraction decomposition)

I am working on the integral of the reciprocal of polynomial such that $\int{\frac{x^{q-1}dx}{x^{p}-ax^{q}+1}}$ where $p$ and $q$ are coprime integers. I tried to solve it by partial fraction ...
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### How do I solve a partial fractions question when $x$ can’t equal any solutions?
How do I solve $A(i)$ and $A(ii)$ when $x$ can't equal $-2/5$ or $1/2$? The only method I’m aware of for partial fractions requires these values to be subbed in and I’m confused. Sorry if the image is ...