Thank you for reporting, we will resolve it shortly
Q.
The value of the definite integral $\int\limits_{ e }^{ e ^{2010}} \frac{1}{ x }\left(1+\frac{1-\ln x }{\ln x \ln \left(\frac{ x }{\ln x }\right)}\right) dx$, equals
Integrals
Solution:
Substituting $\ln x = t$, we get
$\int\limits_1^{2010}\left(1+\frac{1-t}{t(t-(\ln t))}\right) d t=2009+\int\limits_1^{2010-\ln 2010} \frac{-1}{u} d u \quad(u=t-\ln t)$
$=2009-\ln (2010-\ln 2010)$