Consider the equation
$\int_{1}^{e} \frac{{(\log_{e} x)}^{1 / 2}}{x {(a - {(\log_{e} x)}^{3 / 2})}^{2}} d x = 1$ , $a \in (- \infty, 0) \cup (1, \infty)$
Which of the following statements is NOT TRUE?

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sum of the values of a satisfying the above equation is 1

positive integer just greater than the product of values of a satisfying the above equation is 1

An irrational number a satisfies the above equation

exactly one integer value of a satisfies the above the equation

answer is D.

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Detailed Solution

$\begin{array}{l} \int_{1}^{e} \frac{{(\log_{e} x)}^{1 / 2}}{x {(a - {(\log_{e} x)}^{3 / 2})}^{2}} = 1 \\ L e t a - {(\log_{e} x)}^{3 / 2} = t \\ \frac{{(\log_{e} x)}^{1 / 2}}{x} d x = - \frac{2}{3} d t \\ = \frac{2}{3} \int_{a}^{a - 1} \frac{- d t}{t^{2}} = \frac{2}{3} {(\frac{1}{t})}_{a}^{a - 1} = 1 \\ \Rightarrow \frac{2}{3 a (a - 1)} = 1 \\ \Rightarrow 3 a^{2} - 3 a - 2 = 0 \\ \Rightarrow a = \frac{3 \pm \sqrt{33}}{6} \end{array}$