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Q.
The value of the function $f(x)=1+x+\int\limits_1^x\left(\ln ^2 t+2 \ln t\right) d t$, where $f^{\prime}(x)$ vanishes is:
Integrals
Solution:
$f(x)=1+x+\int\limits_1^x\left(\ell n^2 t+2 \ell n t\right) d t$
Differentiate both sides w.r.t. $x$
by using Leibinitz theorem
$ f ^{\prime}( x )=1+\ell n ^2 x +2 \ln x =0$
$(1+\ell nx )^2=0 \therefore \ell nx =-1 $
$ \therefore x =\frac{1}{ e }$