Does this integration have a closed form solution?

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Discussion Overview

The discussion revolves around the integration involved in solving a differential equation related to the dynamics of a system. Participants explore methods to approach the integration, including the application of the binomial theorem.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in solving the integration and seeks intuition on how to approach it.
  • Another participant suggests that the binomial theorem can simplify the integration, indicating that the sum equals 1.
  • Some participants question whether it is valid to directly take the binomial term as 1, expressing concern about the implications of this assumption.
  • A later reply reiterates the application of the binomial theorem, providing the formula for clarification.

Areas of Agreement / Disagreement

There is no consensus on whether the binomial term can be directly taken as 1, as some participants express concerns about this assumption while others support its application.

Contextual Notes

Participants have not resolved the validity of taking the binomial term as 1, and there may be missing assumptions regarding the conditions under which this simplification holds.

anita chandra
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I was trying to solve a differential equation that I defined to study the dynamics of a system. Meanwhile, I encounter integration. The integration is shown in the image below. I tried some solutions but I am failed to get a solution. In one solution, I took "x" common from the denominator terms and then apply a partial method to solve the equation. But that does not work. I request the members of this forum to give me at least an intuition to how can I solve this integration. Thanks a lot.
 

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Hint: By the binomial theorem, the sum is equal to ##1## and thus you have to solve an easy integral.
 
Yes, taking binomial part as 1 can make the solution of equation easy. But my only concern is that can I directly take that term as 1.
 
Last edited:
anita chandra said:
Yes, taking binomial part as 1 can make the solution of equation easy. But my only concern is that can I directly take that term as 1.

Yes, the binomial theorem asserts that

$$(a+b)^n =\sum_{k=0}^n \binom{n}{k} a^k b^{n-k}$$

Apply it and you will be able to conclude.
 
Thanks a lot.
 
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