I've decided to find various points with inputs close to one, using the Desmos Graphing Calculator. You can find what I did here, though you could easily replicate this yourself. Setting the value of a to oscillate between values closer and closer to one, it became clear that the limit goes towards ln(2). I'm not sure of how to go about this problem otherwise, as this is a special integral, but I think that this solution is a good start towards understanding.

Note: If my logic appears to be faulty, or you would like more help, reply to my comment and I will rectify whatever the issue may be.

I would like to know if there is any method to make an estimation on how big it is a number, in terms of orders of magnitude of n! when n is a big number.

(x^2) (2-x)^2 = 1 + 2(2-x)^2 I asked this problem on this page a few months ago, but didn’t get any satisfactory answer. I think there is a trick solution to it

I've decided to find various points with inputs close to one, using the

Desmos Graphing Calculator. You can find what I didhere, though you could easily replicate this yourself. Setting the value of a to oscillate between values closer and closer to one, it became clear that the limit goes towards ln(2). I'm not sure of how to go about this problem otherwise, as this is a special integral, but I think that this solution is a good start towards understanding.Note: If my logic appears to be faulty, or you would like more help, reply to my comment and I will rectify whatever the issue may be.

I am also getting the answer as ln2 using the Taylor series.

Sorry for the delay I got out ln (2) but in Wolfram alpha it tells me that I should leave ln (-2)