If ∫ _ (1)/(3) ^ 3 | _ e x|dx = (m)/(n) _ e ( n ^ 2 e ) , where m...

Let

$I = {∫}_{\dfrac{1}{3}}^{3}\left|\ln x\right|dx$

$⇒ I = {∫}_{\dfrac{1}{3}}^{1}\left|\ln x\right|dx + {∫}_{1}^{3}\left|\ln x\right|dx$

$⇒ I = - {∫}_{\dfrac{1}{3}}^{1}lnxdx + {∫}_{1}^{3}lnxdx$

Since, $\ln x < 0forx \in \left(0, 1\right)$.

$⇒ I = - {\left[xlnx - x\right]}_{\dfrac{1}{3}}^{1} + {\left[xlnx - x\right]}_{1}^{3}$

$⇒ I = - \left[\left(0 - 1\right) - \left(- \dfrac{1}{3}\ln 3 - \dfrac{1}{3}\right)\right] + \left[3\ln 3 - 3 - \left(0 - 1\right)\right]$

$⇒ I = \dfrac{2}{3} - \dfrac{1}{3}\ln 3 + 3\ln 3 - 2$

$⇒ I = \dfrac{8}{3}\ln 3 - \dfrac{4}{3}$

$⇒ I = \dfrac{4}{3}\left(2\ln 3 - 1\right)$

$⇒ I = \dfrac{4}{3}\left(ln{3}^{2} - lne\right)$

$⇒ I = \dfrac{4}{3}\ln \left(\dfrac{{3}^{2}}{e}\right)$

So, $m = 4, n = 3$

Hence, ${m}^{2} + {n}^{2} - 5 = 16 + 9 - 5 = 20$