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Question-40930




Question Number 40930 by scientist last updated on 29/Jul/18
Commented by prakash jain last updated on 29/Jul/18
f_k (x)=e^e^(⋰k times^x )    (d/dx)f_k (x)=f_k (x)(d/dx)f_(k−1) (x)  =f_k (x)f_(k−1) (x)...f_1 (x)  ∫f_k (x)f_(k−1) (x)...f_1 (x)dx=f_k (x)+C  for the given question k=12.
$${f}_{{k}} \left({x}\right)={e}^{{e}^{\iddots{k}\:{times}^{{x}} } } \\ $$$$\frac{{d}}{{dx}}{f}_{{k}} \left({x}\right)={f}_{{k}} \left({x}\right)\frac{{d}}{{dx}}{f}_{{k}−\mathrm{1}} \left({x}\right) \\ $$$$={f}_{{k}} \left({x}\right){f}_{{k}−\mathrm{1}} \left({x}\right)…{f}_{\mathrm{1}} \left({x}\right) \\ $$$$\int{f}_{{k}} \left({x}\right){f}_{{k}−\mathrm{1}} \left({x}\right)…{f}_{\mathrm{1}} \left({x}\right){dx}={f}_{{k}} \left({x}\right)+{C} \\ $$$${for}\:{the}\:{given}\:{question}\:{k}=\mathrm{12}. \\ $$

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