Let f(x)+f(1x)=F(x)  ,where f(x)=∫1xℓnt1+tdt  then F(e)=

F(e)=f(e)+f(1e) =∫1elnt1+tdt+∫11/elnt1+tdt In second integral substitute t=1u =F(e)∫1elnt1+t(1+1t) dt ∫1elntt dt =[(lnt)22]12=12