This function is a parametric curve f(t)=(x(t),y(t)), where
x(t)=lnt/t and y(t)=ln(t+2).
To measure its length we sould first use Pythagorean theorem to find an element of lenght ds (a curve is approximated with a series of small, infinitesimal segments)
ds=sqrt(dx^2+dy^2)
Now because x and y are functions of t we use chain rule
ds=sqrt((dx/dt)^2 dt^2+(dy/dt)^2 dt^2)=sqrt((dx/dt)^2+(dy/dt)^2)dt
The total lenght for t in [1,e] is an integral
s=int_"curve"ds=int_1^e sqrt((dx/dt)^2+(dy/dt)^2)dt
Let's find dx/dt and dy/dt first.
x(t)=lnt*1/t
By product rule
dx/dt=1/t*1/t+lnt*1/t^2=(lnt+1)/t^2
y(t)=ln(t+2)t
By chain rule
dy/dt=1/(t+2)*1
By plugging this in, we get
s=int_1^e sqrt(((lnt+1)/t^2)^2+(1/(t+2))^2)dt
s=int_1^e sqrt((lnt+1)^2/t^4+1/(t+2)^2)dt
Common denominator
s=int_1^e sqrt((t+2)^2(lnt+1)^2+t^4)/(t^2(t+2))dt
s=int_1^e sqrt((t+2)^2ln^2(et)+t^4)/(t^2(t+2))dt
