The rubber band at time t is 1+t, the beetle crawls at time dt 1 dt, and the crawling percentage is 1/( 1+t)dt.

So ∫(0 to t)1(1+t) dt =1,

The integral is ln( 1+T)= 1, so T=e- 1.

The length of the rubber band is e-1+1= e m.