Finding the Limit of ∫N^2/[1+(NX)^2]^2 as N→∞

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In summary, the conversation discusses how to find the limit of an integral as N goes to infinity. The integral in question is ∫{N^2/[1+ (NX)^2]^2}dX, 0<X<1 and the method involves using substitution and splitting fractions. The speaker also mentions proving the integral and taking the limit either inside or outside the integral.
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t.t.h8701
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How to find the limit of ∫{N^2/[1+ (NX)^2]^2}dX, 0<X<1; as N goes to infinite?
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So I checked symbolically that
[tex] \int \frac{1}{(1 + x^2)^2} \, dx = \tfrac12\left( \frac{x}{1 + x^2} + \operatorname{arctan}(x) \right). [/tex]
Of course the arctan comes from an integration of 1/(1 + x^2) so although I haven't fully done the calculation splitting fractions seems like the way to go. I suspected something like
[tex] \frac{1}{(1 + x^2)^2} = \frac{1}{1 + x^2} - \frac{x^2}{(1 + x^2)^2} [/tex]
but don't kill me if I'm wrong.

Once you have proven that integral, you can easily find the right substitution and take the limit (say, the integral is continuous, converges, yaddayadda, so we can take the limit in- or outside the integral as we like, etc. - you get the point).
 

Related to Finding the Limit of ∫N^2/[1+(NX)^2]^2 as N→∞

1. What is the purpose of finding the limit of this integral as N approaches infinity?

The limit of this integral helps us understand the behavior of the function as N gets larger and larger. It can also help us determine if the integral has a finite or infinite value.

2. How do you solve for the limit of this integral as N approaches infinity?

To solve for the limit, we use the concept of the Fundamental Theorem of Calculus and take the derivative of the function inside the integral. We then substitute N=∞ into the resulting function and solve for the limit.

3. What is the significance of the term "∞" in the limit of this integral?

The term "∞", or infinity, represents the idea of an unbounded value, meaning that the function will continue to increase without bound as N gets larger and larger. It can also indicate an infinite area under the curve of the function.

4. Are there any special cases or exceptions when finding the limit of this integral as N approaches infinity?

Yes, there are certain cases where the limit may not exist or may require additional techniques to solve. For example, if the function inside the integral is oscillating or if there are discontinuities, the limit may not exist. In these cases, we may need to use techniques such as the Squeeze Theorem to find the limit.

5. How can knowing the limit of this integral as N approaches infinity be applied in real-world situations?

The concept of limits is used in various fields such as physics, engineering, and economics to model and analyze real-world phenomena. Knowing the limit of this integral can help us understand the behavior of systems that involve growth or decay over time, such as population growth or the decay of radioactive substances.

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