Impossible integral?

[FrederikDan]

I'm trying to determine the "doubling time" of the following function as a function of T. I have tried converting the function to the integral of an infinite sum, but that didn't seem to clarify things. I am aware, that there most likely is no analytical solution to the integral itself, however, I was wondering if there might be a way to determine an analytical solution for its doubling time as a function of T.

 

[FrederikDan]

 

[Cubist]

I think that the symbolused within the OP's image means [imath]\sqrt{x^3}[/imath] (please correct me if I'm wrong)

The following step in your work looks dubious to me...



It appears like you're trying to raise all the individual elements of a sum to the same power. Here's an analogy...
[math]3^2+4^2 = 5^2 + 0^2[/math][math]\sqrt{3^2}+\sqrt{4^2} \ne \sqrt{5^2} + \sqrt{0^2}[/math][math]3+4 \ne 5[/math]