Question

$\int \frac{1}{x\log x\left[\log \right(\log x\left)\right]}dx=$

• A. $\log |\log \left(\log x\right)|+c$
• B. $\log |\log x|+c$
• C. $-\log |\log x|+c$
• D. $-\log |\log \left(\log x\right)|+c$

Answer 1

The correct option is A $\log |\log \left(\log x\right)|+c$

$\int \frac{1}{xlogx\left[log\right(logx\left)\right]}dx$

$dlogx={}^1{/}_x.dx$

$\int \frac{dlogx}{logx\left[log\right(logx\left)\right]}$

log x = t

$\int \frac{dt}{t}\left[logt\right]$

$=\int \frac{\left[dlog\right(t\left)\right]}{log\left(t\right)}$

$logt=p$

$=logp+c$

$=log\left(logt\right)+c$

=$log\left|log\right(logx\left)\right|+c$

$\int \frac{1}{xlogx\left[log\right(logx\left)\right]}=log\left|log\right(logx\left)\right|+c$