Question

Evaluate ${\int }_{10}^{\infty }\frac{dx}{x\ln x}$

• A. $1$
• B. $-10$
• C. $\infty$
• D. Undefined

Answer 1

Answer: (C)

$\infty$

Given : ${\int }_{10}^{\infty }\frac{dx}{x\ln x}$

${\int }_{10}^{\infty }\frac{dx}{x\ln x}=\stackrel{lim}{c\to \infty }{\int }_{10}^c\frac{dx}{x\ln x}$

$\ln x=tdx=dt$

${\int }_{10}^{\infty }\frac{dx}{x\ln x}=\stackrel{lim}{c\to \infty }{\int }_{\ln 10}^{\ln c}\frac{dt}{\left(t\right)}{\int }_{10}^{\infty }\frac{dx}{x\ln x}=\stackrel{lim}{c\to \infty }{\left|\ln t\right|}_{10}^{\ln c}{\int }_{10}^{\infty }\frac{dx}{x\ln x}=\stackrel{lim}{c\to \infty }\ln \left|\ln \left|c\right|\right|-\ln \left|\ln \left|10\right|\right|=\infty$

Hence the correct answer is $\infty$