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#
