# Questions tagged [partial-fractions]

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

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### Sum of finite series using partial fraction

I'm quite stuck with the following problem. I have seen on this forum that there is already an answer for the infinite sum to the problem but I can't seem to find how to find the sum for a finite ...
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### Formula for $kth$ convergent of a simple continued fraction

Let $\frac{p}{k}$ be a rational number with partial fraction decomposition $C_n=[a_0; a_1,...a_n]$ and for $k<n$ let $C_k=[a_0;a_1,...a_k]$ be the partial fraction decomposition of the $kth$ ...
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### How do I solve the partial fractions $\frac{x+1}{(x^2+1)(x^2+x+1)}$? [closed]

I have the following problem: How can I break this fraction down into a simple fraction $\frac{x^3+x^2+x+1}{(x^2+1)^2(x^2+x+1)}$ Can you help me? I applied the algorithm, but I couldn't solve it. ...
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### Partial Fraction Decomposition of $\int_{0}^{\infty} \frac{e^{-\frac{w}{s}}}{\left(mw+A\right) \left( mw+ B\right)^{L}}dw$

I'm sorry the title not allow more than 150 characters, I couldn't put a full-length integral equation I tried to simplify the equation to decrease less than 150 char. Here is the below full of the ...
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Why is there Bx+c term when we try to split partial fraction with irreducible quadratic? Eg: $$\frac{1}{x(x^2+1)} = \frac{A}{x} + \frac{Bx+C}{x^2+1}$$ I think that splitting partial fraction is ...
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### Could $\int\frac{1}{x^{N+1}(x-1)}dx$ be solved analytically?

I am trying to solve this integral: $$\int\frac{1}{x^{N+1}(x-1)}dx$$ I have tried integration by partial fraction, substitution and by parts. But, I can't solve it. So, I would like to ask could this ...
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### Nice integral $\int_{0}^{\infty}\ln\Big(\frac{x^3-x^2-x+1}{x^3+x^2+x+1}\Big)\frac{1}{x}dx=-\frac{3\pi^2}{4}$

Last integral of the day : $$\int_{0}^{\infty}\ln\Big(\frac{x^3-x^2-x+1}{x^3+x^2+x+1}\Big)\frac{1}{x}dx=-\frac{3\pi^2}{4}$$ I have tried integration by parts and some obvious substitution but I ...
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### Let $f(x) = \frac{-2x+4}{(x^2+1)(x-1)^2}$. Express the function $f(x)$ as a sum of partial fractions.

I have gotten $(-2x+4) = A(x-1)^2 + B((x^2+1))$ But after letting $x = 1$, $B = 1$, I couldn't find $A$. Any help is appreciated. Thank you
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### Partial Fractions $\frac{2}{s^2+4} = \frac{As+B}{s^2+4}$

I have this relatively simple partial fraction $$\frac{2}{s^2+4} = \frac{As+B}{s^2+4}$$ I multiply each side by $(s^2+4)$ and all that remains is $2 = As + B$. Then can I match the coefficients up ...
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### How can I evaluate this integral by partial decomposition?

How can I evaluate this integral, if the denominator has a quadratic factors e.i-($b^{2}-4ac<0$) $$\int\frac{xdx}{x^{2}+6x+13}$$ by partial decomposition?
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### Compute $\int\frac {x^2}{x^4+1}dx$ via partial fractions

I am trying to solve it with "partial fractions" $$\frac {x^2}{x^4+1}=\frac{x^2}{(x^2+x\sqrt{2}+1)(x^2-x\sqrt{2}+1)}=\frac{Ax+B}{(x^2+x\sqrt{2}+1)}+\frac{Cx+D}{(x^2-x\sqrt{2}+1)}$$ and I get the ...
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### Row Echelon, Partial Fractions, and Numerator Coefficients

I am trying to get the numerator values for the partial fraction decomposition of: $$\dfrac{x^2+1}{x(x-1)(x+1)(x-4)}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1}+\frac{D}{x-4}$$ I really started hitting ...
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### How the partial fraction decomposition works for finding this Inverse Laplace Transform?

I've been working to find inverse Laplace transform for the following : $$\frac{A}{(s-a)(s-r_1)(s-r_2)}$$ However, I'm getting stuck on the partial fraction decomposition. When I run the ...
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### How to get from $(Ae^t.t)/(t+1)^2$ to $Ae^t/1+t - Ae^t/(1+t)^2$?

I've been told that $(Ae^t.t)/(t+1)^2$ $=$ $Ae^t/1+t - Ae^t/(1+t)^2$ but im not sure how. I thought of using partial fractions but im not quite sure which case this would be. Would appreciate the ...
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### Why is $\sum_{i=1}^n\frac n{n-i+1}=n\sum_{i=1}^n\frac1i$?

While reading through my (algorithms and) probability script, I have seen this equality for calculating the first moment of the coupon-collector problem. However, I don't quite see how the sum of the ...
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### Partial fractions decomposition. Why $cx+d$ instead of $cx$ for the numerator of $(x^2+2$).

I understand that the aim of partial fractions decomp. is simply to reach (an) integrable functions, but then I have trouble wrapping my head around why you cannot make the numerator of something like ...
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### Partial Frac Exp

Someone can explain step by step how i can manage this P(s) in order to apply the inverse laplace transform ? $P(s) = \frac{s^3+5s^2+3}{s^2(s^2-3s-18)}$ I tried this way of fractioning but it ...
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### If the value of integral in the image below is π then what is the value of y?

I could not simplify $$\int_0^1 \sqrt{-1 + \sqrt{\frac{1+y}{x} - y}}\ dx$$ I tried integration it in an online integrator but trust me the result is seriously daunting to be back traced to $\pi$ as ...
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### Finding the value of $\sum_{n=1}^\infty\frac{n^2}{(n+1)(n+2)(n+3)(n+4)}$ [duplicate]

Problem_ Find the value of $$\sum_{n=1}^\infty\frac{n^2}{(n+1)(n+2)(n+3)(n+4)}$$ It seems like I have to use the partial sum in order to get the exact value. But making it into the partial ...
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### What's the best way to solve $1 = A(x^2+1) + (Bx+C)(x+1)$

What's the best way to solve $$1 = A(x^2+1) + (Bx+C)(x+1)$$ I let $x=-1$ and got $A=\frac{1}{2}$ But what sub is ideal to find B&C This gets messy quick, I think. Instead, I started over, and I ...
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### integration of. $\int \frac{x}{x^3-3x+2}$

I am trying to integrate : $\Large \int \frac{x}{x^3-3x+2}dx$ I decomposed the fraction and got : $\Large \frac{x}{x^3-3x+2} = \frac {x}{(x-1)^2(x+2)}$ Then I tried to get two different ...
### show that the portion of the partial fraction expansion of $P(x)/Q(x)$ corresponding to the factor $x-r$ Is $A/(x-r)$ where $A=P(r)/Q’(r)$
If $r$ Is a non-repeated root of $Q(x)$, show that the portion of the partial fraction expansion of $P(x)/Q(x)$ corresponding to the factor $x-r$ Is $A/(x-r)$ where $A=P(r)/Q’(r)$ I’m looking for ...