# What is the arclength of ((e^tlnt)/t,t/(e^tlnt)) on t in [2,4]?

Dec 25, 2015

$f \left(t\right) = \frac{{e}^{t} \ln \left(t\right)}{t}$

$g \left(t\right) = \frac{t}{{e}^{t} \ln \left(t\right)}$

$\frac{\mathrm{df}}{\mathrm{dt}} = \frac{{e}^{t} \left(\left(t - 1\right) \ln \left(t\right) + 1\right)}{t} ^ 2$

$\frac{\mathrm{dg}}{\mathrm{dt}} = \frac{{e}^{- t} \left(\ln \left(t\right) \left(1 - t\right) - 1\right)}{\ln} ^ 2 \left(t\right)$

$\Gamma \left(t\right) = {\int}_{2}^{4} \sqrt{{\left(\frac{\mathrm{df}}{\mathrm{dt}}\right)}^{2} + {\left(\frac{\mathrm{dg}}{\mathrm{dt}}\right)}^{2}} \mathrm{dt}$

$\Gamma \left(t\right) = {\int}_{2}^{4} \sqrt{{\left(\frac{{e}^{t} \left(\left(t - 1\right) \ln \left(t\right) + 1\right)}{t} ^ 2\right)}^{2} + {\left(\frac{{e}^{- t} \left(\ln \left(t\right) \left(1 - t\right) - 1\right)}{\ln} ^ 2 \left(t\right)\right)}^{2}}$

Which is $\approx 16.371300234854$

Note : Of course integral is not defined