A Does this integration have a closed form solution?

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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Math_QED

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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:

Math_QED

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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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