Question

Consider the equation
$\int\limits_1^e \frac{\left(\log _{ e } x \right)^{1 / 2}}{x\left(a-\left(\log _{ e } x \right)^{3 / 2}\right)^2} dx =1, \quad a \in(-\infty, 0) \cup(1, \infty) .$
Which of the following statements is/are TRUE ?

• A. No $a$ satisfies the above equation
• B. An integer $a$ satisfies the above equation
• C. An irrational number $a$ satisfies the above equation
• D. More than one $a$ satisfy the above equation

Answer 1

Answer: (C), (D)

$\int\limits_1^e \frac{\left(\log _c x\right)^{1 / 2}}{x\left(a-\left(\log _c x\right)^{3 / 2}\right)^2}=1$
Let $a-\left(\log _c x \right)^{3 / 2}= t$
$ \frac{\left(\log _e x\right)^{1 / 2}}{x} d x=-\frac{2}{3} d t $
$ =\frac{2}{3} \int\limits_a^{a-1} \frac{-d t}{t^2}=\frac{2}{3}\left(\frac{1}{t}\right)_a^{a-1}=1 $
$ \frac{2}{3 a(a-1)}=1$
$3 a^2-3 a-2=0 $
$ a=\frac{3 \pm \sqrt{33}}{6}$