A useful approximation!

$approximation\quad of\quad \int _{ \frac { 1 }{ e } }^{ e }{ { x }^{ \frac { 1 }{ x } } } dx\\ \int _{ \frac { 1 }{ e } }^{ e }{ { x }^{ \frac { 1 }{ x } } } dx\quad =\quad \int _{ \frac { 1 }{ e } }^{ e }{ { e }^{ \frac { logx }{ x } } } dx\quad =\quad I\\ Since\quad we\quad want\quad to\quad approximate\quad the\quad value\quad of\quad I,\\ we\quad can\quad use\quad taylor\quad series\quad of\quad expansion\quad of\quad { e }^{ x }\\ I\quad =\quad \int _{ \frac { 1 }{ e } }^{ e }{ \left( 1+\frac { \frac { logx }{ x } }{ 1! } +\frac { { \left( \frac { logx }{ x } \right) }^{ 2 } }{ 2! } +\frac { { \left( \frac { logx }{ x } \right) }^{ 3 } }{ 3! } +... \right) } dx\\ on\quad splitting\quad the\quad integrals\quad and\quad solving\quad them,\quad we\quad get\\ I\quad \approx \quad 2.65861$