Question Number 43803 by vc526 last updated on 15/Sep/18
$$\infty \\ $$$$\int\:\:\:\:\frac{\mathrm{1}}{{x}\:\mathrm{l}{n}\:{x}}{dx}\:\:\:=\: \\ $$$$\mathrm{0} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 15/Sep/18
$${t}={lnx} \\ $$$${dt}=\frac{{dx}}{{x}} \\ $$$$\int_{−\infty} ^{\infty} \frac{{dt}}{{t}} \\ $$$$\underset{{a}\rightarrow−\infty} {\mathrm{lim}}\int_{{a}} ^{\infty} \frac{{dt}}{{t}} \\ $$$$=\underset{{a}\rightarrow−\infty} {\mathrm{lim}}\mid{lnt}\mid_{{a}} ^{\infty} \\ $$$$=\underset{{a}\rightarrow−\infty} {\mathrm{lim}}\left\{{ln}\infty−{lna}\right\} \\ $$$$=\underset{{a}\rightarrow−\infty} {\mathrm{lim}}\:\left\{\infty−{lna}\right\} \\ $$$$ \\ $$