Question

Consider the equation
$\underset{1}{\overset{e}{\int }}\frac{{\left({\log }_{e}x\right)}^{1/2}}{x{\left(a-{\left({\log }_{e}x\right)}^{3/2}\right)}^2}dx=1,a\in (-\infty ,0)\cup (1,\infty ).$
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)

$\underset{1}{\overset{e}{\int }}\frac{{\left({\log }_{c}x\right)}^{1/2}}{x{\left(a-{\left({\log }_{c}x\right)}^{3/2}\right)}^2}=1$
Let $a-{\left({\log }_{c}x\right)}^{3/2}=t$
$\frac{{\left({\log }_{e}x\right)}^{1/2}}{x}dx=-\frac{2}{3}dt$
$=\frac{2}{3}\underset{a}{\overset{a-1}{\int }}\frac{-dt}{t^2}=\frac{2}{3}{\left(\frac{1}{t}\right)}_a^{a-1}=1$
$\frac{2}{3a(a-1)}=1$
$3a^2-3a-2=0$
$a=\frac{3\pm \sqrt{33}}{6}$