Here's as close as I can get via Mathematica.  First, I just simplify the integrand once for all.  Having Simplify in the definition of a function could be really slow.  Edit: I added  the unsimplified versions of the OP's functions, including a substitution of Boole for If, which I omitted to include in the original answer.

u[n_, x_] := Boole[0 &lt;= x &lt;= L] Sqrt[2/L] Sin[n π x/L];
φ[n_, k_] := FourierTransform[u[n, x], x, k, FourierParameters -&gt; {0, -1}];
η[n_, k_] := φ[n, k]\[Conjugate] φ[n, k];

integrand =
  FullSimplify[k^2 \[Eta][n, k],
               {n \[Element] Integers, n &gt; 0, k \[Element] Reals,L &gt; 0}]
(* -((2 k^2 L n^2 \[Pi] (-1 + (-1)^n Cos[k L]))/(k^2 L^2 - n^2 \[Pi]^2)^2) *)


Then, Integrate gets real close, if you make a substitution k -&gt; n k.  Since the differential of n k is n dk, you have to multiply the Integrate below by n.

int = n Integrate[integrand /. k -&gt; n k, {k, -Infinity, Infinity}];
Simplify[int, {n \[Element] Integers, n &gt; 0}]

(* ConditionalExpression[
    (L n^2 Pi^2)/Abs[L]^3, 
    L \[Element] Reals &amp;&amp; 3 Arg[-L^2] &lt;= 2 \[Pi] &amp;&amp; (Re[1/L^2] &lt;= 0 || 1/L^2 \[NotElement] Reals)] *)

Reduce[Last[%]]
(* False *)


The only problem is that the condition in the ConditionalExpression is a contradiction.

Calculate integral for arbitrary parameter n in infinite square well problem
Given the comments regarding the cost of true hardware RAID in some of my searches like here or here, it would appear that $500 solutions like Synology DS413, Netgear ReadyNAS Ultra 4, QNAP TS-419P must use software-based RAID.  True?  Are they using a hybrid hardware-software combo?

All manufacturers appear to offer some proprietary version of expandable RAID (Netgear X-RAID, Synology Hybrid RAID, etc).  I believe these boxes run Linux inside, are these proven Linux software RAID algorithms that the manufacturers are branding, or truly proprietary?  Any less reliable than tried-and-true RAID-1, 5, 10, etc?

I would just like to understand the internals of these machines a little better before trusting them with terabytes of important data.  Thanks.
