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

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The discussion focuses on finding the limit of the integral ∫N^2/[1+(NX)^2]^2 dX as N approaches infinity, specifically for the range 0 < X < 1. Participants confirm that the integral ∫1/(1+x^2)^2 dx evaluates to (1/2)(x/(1+x^2) + arctan(x)). The method involves splitting fractions to simplify the integral, and once established, the limit can be taken either inside or outside the integral due to its continuity and convergence properties.

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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?
Thanks
 
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So I checked symbolically that
\int \frac{1}{(1 + x^2)^2} \, dx = \tfrac12\left( \frac{x}{1 + x^2} + \operatorname{arctan}(x) \right).
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
\frac{1}{(1 + x^2)^2} = \frac{1}{1 + x^2} - \frac{x^2}{(1 + x^2)^2}
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).
 

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