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

[t.t.h8701]

How to find the limit of ∫{N^2/[1+ (NX)^2]^2}dX, 0<X<1; as N goes to infinite?
Thanks

 

[CompuChip]

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).