Question

The value of definite integral $\int\limits_1^{ e } \frac{ dx }{\sqrt{ x ^2 \ln x +( x \ln x )^2}}$ 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 =\int\limits_1^{ e } \frac{ dx }{ x \sqrt{\ln x (1+\ln x )}} $
Put $1+\ln x = t ^2$
$I =\int\limits_1^{\sqrt{2}} \frac{2 tdt }{ t \sqrt{ t ^2-1}}=2 \ln (\sqrt{2}+1)=\ln (3+2 \sqrt{2})$.