Define

The original question takes the argument a=2.

For brevity, we will suppress the first two arguments⁰, as it turns out they are irrelevant to the final answer.

(This is a thing in higher mathematics lmao get used to it)

The original question is equivalent¹ to:

It is clear that ψ(u,v) = ψ(v,u), from the definition, so we can collapse the first two terms together.

Break off the +1 term and ignore it for now, as that is what gives the final answer.

Using the substitution x = t², the integral becomes:

Using the boundary preserving transformation t→1-t, this becomes:

Use symmetry in u,v, once again, to reorder the arguments to be the same as before.

Take the average of this and the previous step, to obtain:

We consider only the last two terms for now.

Define:

By the same boundary preserving transformation as before, it is obvious that A = B.

Use the scaling transformation t→2t:

Use the same "boundary preserving transformation" to change the interval [0,½] → [½,1]

Finally, we have:

Returning to the step before we defined A and B, we obtain:

The +1 term is trivially integrated to obtain 1, which is the final answer.

⁰ ψ(u,v) = ψ(a,k,u,v)

¹ left as an exercise to the reader