Finding the arc length of a path between two points

First note that 
\[c(1)=(1,0,2)    and   c(2)=(4,\ln 2, 4). \]
Also $c'(t)=(2t, \frac{1}{t}, 2)$. So 
\[|c'(t)|=\sqrt{4t^2+\frac{1}{t^2}+4}=\sqrt{(2t+\frac{1}{t})^2}=2t+\frac{1}{t},   t>0.\]
Thus the arclength between the two points is
\[L=\int_1^2|c'(t)| dt =\int_1^2 2t+\frac{1}{t} dt= t^2+\ln t |_1^2=(4+\ln 2)-(1+\ln 1)=3+\ln 2.\]