# Questions tagged [partial-fractions]

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

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### How to Evaluate the sum $\sum_{k=1}^n \frac{2 k+1}{k(k+1)}$ by the partial fractions $\frac{1}{k}-\frac{1}{k+1}$ [closed]

I am trying to find the answer to this sum: $$\sum_{k=1}^n \frac{2 k+1}{k(k+1)}$$ but I need to find it through a partial fraction method: $$\frac{1}{k}-\frac{1}{k+1}$$ How di I do that? I also ...
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### How does partial fraction expansion generalize to fractions of integers? Why is it not unique, in that case?

The Wikipedia page for partial fraction expansion mentions that it can be generalised to "regular" fractions, i.e. fractions of integers: https://en.wikipedia.org/wiki/...
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### How do I complete this partial fraction

I am trying to take perform a partial fraction decomposition on the following function but something is not correct.$$Y(x)=\frac{(5x+10)(\cos x)}{x^2(x+9+i)(x+9-i)}$$ that I am trying to take the ...
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### Partial fractions of equal degree [closed]

I have the integral and I thought I was supposed to use partial fraction decomposition, but I was not getting the correct answer. $$\int\frac{x^3 + 3x^2 + 2x + 1}{x^3 + x}\mathrm dx$$
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### Simplify Using the Fundamental Theorem of Calculus

I am just starting my Partial Differential Equations course. I am looking at the steps to derive the heat equation for a rod, and there is one step I can't figure out. It says that the following can ...
1 vote
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### Why is that integral written as the product of 2 other integrals?

I am a student and just learning about integrals. **Could you please explain to me how the highlighted part of the equation was derived?** I can't quite understand how was the c/a term brought of the ...
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### Why can we treat the numerator as a constant in partial fractions?

I'm learning the method of partial fraction decomposition as a 'useful dodge' (Silvanus Thompson, Calculus Made Easy) for calculus problems, but I'm not quite following the reasoning. According to ...
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### Find particular solution for $(D^2+1)y=e^{a \cos x}$, where a is an arbitary constant.

I tried solving the problem as in the image by taking partial integrals and also by series expansion, but it is becoming more complex to continue and find a close form of it. Please solve it.
1 vote
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### Which integrating technique should I use?

Just some context: In the mathematical course, I have undertaken this year, I've just learnt how to integrate using partial fractions, substitution(not trig though, just a variable) and integrating ...
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### Taking the partial fraction when the variable is exponential

I'm trying to solve the following integral: $\int\frac{dx}{2^x+3}$ and here's what I've done so far: Substituting $t=2^x$ we have $dt=2^x\ln 2 dx\implies \frac{dt}{t}=\ln2dx$ Now, I have to take the ...
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