Question

$\underset{1}{\overset{e}{\int }}(\frac{1}{\sqrt{x\ln x}}+\sqrt{\frac{\ln x}{x}})dx$ equals

• A. $\sqrt{e}$
• B. $2e$
• C. $2\sqrt{e}$
• D. $\sqrt{2e}$

Answer 1

The correct option is C $2\sqrt{e}$

$I=\underset{1}{\overset{e}{\int }}\frac{1+\ln x}{\sqrt{x\ln x}}dx$
put $x\ln x=t^2\Rightarrow (\ln x+1)dx=2tdt$
$I=\underset{0}{\overset{\sqrt{e}}{\int }}\frac{2tdt}{t}=2\sqrt{e}$