Question

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

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

Answer 1

Answer: (A), (B)

An irrational number $a$ satisfies the above equation

Let $I=\underset{1}{\overset{e}{\int }}\frac{(\ln x{)}^{1/2}}{x(a-(\ln x{)}^{3/2}{)}^2}dx$
put $a-(\ln x{)}^{3/2}=t$
$⟹-\frac{3}{2}(\ln x{)}^{1/2}\cdot \frac{1}{x}dx=dt$
$\therefore I=\underset{a}{\overset{a-1}{\int }}\frac{\left(\frac{-2}{3}\right)dt}{t^2}$
$={\left[\left(\frac{-2}{3}\right)\frac{t^{-2+1}}{-2+1}\right]}_a^{a-1}$
$=\frac{2}{3t}{|}_a^{a-1}=\frac{2}{3}(\frac{1}{a-1}-\frac{1}{a})$
$\therefore I=\left(\frac{2}{3}\right)\frac{1}{a(a-1)}=1$
$⟹2=3a^2-3a$
$⟹3a^2-3a-2=0$
$⟹a=\frac{3\pm \sqrt{9-4\left(3\right)(-2)}}{6}$
$a=\frac{3+\sqrt{33}}{6},\frac{3-\sqrt{33}}{6}$