Question

The value of definite integral $\underset{1}{\overset{e}{\int }}\frac{dx}{\sqrt{x^2\ln x+(x\ln x{)}^2}}$ is

• A. $\ln (3+2\sqrt{2})$
• B. $\ln (\sqrt{2}+1)$
• C. $\ln (2\sqrt{2}+1)$
• D. $\ln (3\sqrt{2}+2)$

Answer 1

Answer: (A)

$I=\underset{1}{\overset{e}{\int }}\frac{dx}{x\sqrt{\ln x(1+\ln x)}}$
Put $1+\ln x=t^2$
$I=\underset{1}{\overset{\sqrt{2}}{\int }}\frac{2tdt}{t\sqrt{t^2-1}}=2\ln (\sqrt{2}+1)=\ln (3+2\sqrt{2})$.