$ f \left( e ^3\right)=\int\limits_1^{ e ^3} \frac{\ell nt }{\ln 10(1+ t )} dt \ldots . . .(1) $
$ f (\alpha)=\int\limits_1^a \frac{\ell nt }{(\ln 10)(1+ t )} dt $
$ t =\frac{1}{ x } \Rightarrow x =\frac{1}{ t }$
$ dt =\frac{-1}{ x ^2} dx $
$=\int\limits_1^{\frac{1}{\alpha}} \frac{-\ell \ln x}{(\ell \ln 10)\left(1+\frac{1}{x}\right)}\left(-\frac{1}{x^2}\right) d x $
$ f(\alpha)=\frac{1}{\ell \ln 10} \int\limits_1^{\frac{1}{\alpha}} \frac{\ell nx }{ x ( x +1)} dx$
$ f \left( e ^{-3}\right)=\frac{1}{\ell \ln 10} \int\limits_1^{c^3} \frac{\ell nt }{ t ( t +1)} dt \ldots \ldots \ldots$
Add (1) & (2)
$f\left(e^3\right)+f\left(e^{-3}\right)$
$ =\left(\frac{1}{\ell n 10}\right) \int\limits_1^{ c ^3} \frac{\ell nt }{(1+ t )}\left[1+\frac{1}{ t }\right] dt $
$ =\left(\frac{1}{\ell n 10}\right) \int\limits_1^3 \frac{\ell nt }{ t } dt $
$ \ell nt = r $
$ \frac{ dt }{ t }= dr $
$ =\frac{1}{\ell n 10} \int\limits_0^3 rdr $
$ =\left.\left(\frac{1}{\ell n 10}\right)\left(\frac{ r ^2}{2}\right)\right|_0 ^3 $
$=\left(\frac{1}{\log 10}\right)\left(\frac{9}{2}\right)$
$ =\frac{9}{2 \log _{ e } 10}$