Consider the equation ∫ _ 1 ^ e ( _ e x) ^ (1)/(2) x (a - ( _ e x)...

Let $I = {∫}_{1}^{e}\dfrac{{\left(\ln x\right)}^{\dfrac{1}{2}}dx}{x{\left(a - {\left(\ln x\right)}^{\dfrac{3}{2}}\right)}^{2}}$

Now put $a - \left(\ln x\right)\dfrac{3}{2} = t$

$⇒ - \dfrac{3}{2}\left(\ln x\right)\dfrac{1}{2} \cdot \dfrac{1}{x}dx = dt$

So, the integral becomes $I = {∫}_{a}^{a - 1}\dfrac{\left(- \dfrac{2}{3}\right)dt}{{t}^{2}}$

$⇒ I = \left(- \dfrac{2}{3}\right){\dfrac{{t}^{- 2 + 1}}{- 2 + 1}}_{a}^{a - 1}$

$⇒ I = {\dfrac{2}{3t}}_{a}^{a - 1}$

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

Now given $I = \left(\dfrac{2}{3}\right)\dfrac{1}{a\left(a - 1\right)} = 1$

$⇒ 2 = 3{a}^{2} - 3a$

$⇒ 3{a}^{2} - 3a - 2 = 0$

$⇒ a = \dfrac{3 \pm \sqrt{9 - 4\left(3\right)\left(- 2\right)}}{6}$

So, $a = \dfrac{3 + \sqrt{33}}{6}, \dfrac{3 - \sqrt{33}}{6}$