Does this integration have a closed form solution?

In summary, the conversation discusses the speaker's attempt to solve a differential equation and encountering integration. They tried various solutions, including taking "x" common and applying a partial method, but were unsuccessful. Another member suggests using the binomial theorem to simplify the integration, which the speaker acknowledges can make the solution easier. However, they are unsure if they can directly take the binomial term as 1. The conversation ends with the suggestion to apply the binomial theorem to solve the integration.
  • #1
anita chandra
3
1
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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  • #2
Hint: By the binomial theorem, the sum is equal to ##1## and thus you have to solve an easy integral.
 
  • #3
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.
 
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  • #4
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.
 
  • #5
Thanks a lot.
 
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Likes member 587159

1. What is a closed form solution in integration?

A closed form solution in integration refers to an analytical or algebraic expression that can be used to directly calculate the definite integral of a function without using numerical methods. In other words, it is a formula that gives you the exact solution to an integration problem.

2. How do you know if an integration has a closed form solution?

There is no definitive way to know if an integration has a closed form solution. However, certain characteristics of the integrand, such as its complexity and the types of functions involved, can give an indication of whether a closed form solution exists. In general, simpler and more common functions are more likely to have closed form solutions.

3. What are some examples of integrations with closed form solutions?

Some common examples of integrations with closed form solutions include polynomial functions, trigonometric functions, exponential functions, and logarithmic functions. For instance, the definite integral of x^2 from 0 to 1 has a closed form solution of 1/3, while the definite integral of sin(x) from 0 to pi/2 has a closed form solution of 1.

4. Can a complex function have a closed form solution?

Yes, a complex function can have a closed form solution. However, the complexity of the function may make it difficult to find the closed form solution, or it may not be possible to find one at all. In these cases, numerical methods can be used to approximate the definite integral.

5. How do closed form solutions in integration benefit scientists?

Closed form solutions in integration allow scientists to quickly and accurately calculate definite integrals without having to rely on numerical methods, which can be time-consuming and less precise. This can be especially useful in fields such as physics and engineering, where integrals are commonly used to solve problems and make predictions.

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