Integral of (W(1/t))² Geometrized

Tail piece:

Notice how the left bound is that x = W(1/m) line, and there is no right bound, that means our final integral is this:

...Which then evaluates like this:

...I will also apply a Lambert W identity here (This will end up simplifying.):

We can then sum those terms, and algebraically simplify:

Other integral...

"The total area of the orange triangle scales linearly according to what is added to m in the coefficient here, in this case, 1."

^^^Using this statement from earlier, we can replace the 1 in that (m + 1) coefficient with a differential and integrate the product of said differential and the formula for the area of the orange triangle. Doing this we obtain another integral that is equal to the red area:

We can then multiply by 2 to make the integrand of that aforementioned integral a little bit cleaner: