Question Number 40930 by scientist last updated on 29/Jul/18
Commented by prakash jain last updated on 29/Jul/18
$${f}_{{k}} \left({x}\right)={e}^{{e}^{\iddots{k}\:{times}^{{x}} } } \\ $$$$\frac{{d}}{{dx}}{f}_{{k}} \left({x}\right)={f}_{{k}} \left({x}\right)\frac{{d}}{{dx}}{f}_{{k}−\mathrm{1}} \left({x}\right) \\ $$$$={f}_{{k}} \left({x}\right){f}_{{k}−\mathrm{1}} \left({x}\right)...{f}_{\mathrm{1}} \left({x}\right) \\ $$$$\int{f}_{{k}} \left({x}\right){f}_{{k}−\mathrm{1}} \left({x}\right)...{f}_{\mathrm{1}} \left({x}\right){dx}={f}_{{k}} \left({x}\right)+{C} \\ $$$${for}\:{the}\:{given}\:{question}\:{k}=\mathrm{12}. \\ $$