# Integrate : $\int a^{\large a^{\Large a^{\Large a^{x}}}}.a^{\large a^{\Large a^{x}}}.a^{a^{x}}.a^{x}$$.dx (a)\;\frac{a^{\large a^{\Large a^{\Large a^{x}}}}}{(\log _e a)^4}+c\qquad(b)\;\frac{a^{\large a^{\Large a^{x}}}}{(\log _e a)^4}+c\qquad(c)\;\frac{a^{\large a^{\Large a^{\Large a^{x}}}}}{(\log _e a)^2}+c\qquad (d)\;None ## 1 Answer a^{\large a^{\Large a^{x}}}=t differentiate with x a^{\large a^{\Large a^x}}.\log _e a .a^{\large a^{x}}. \log _e a. a^x. \log _ea. dx =dt => a^{\large a^{\Large a^x}}.a^{\large a^{x}}.a^x( \log _ea)^3. dx =dt => a^{\large a^{\Large a^x}}.a^{\large a^{x}}.a^x( \log _ea)^3. dx =dt => a^{\large a^{\Large a^x}}.a^{\large a^{x}}.a^x. dx =\large\frac{1}{(\log _ea)^3}$$dt$
By putting values we get,
$\qquad= \int a^t \times \large\frac{1}{(\log _ea)^3}$$.dt \qquad= \int a^t . \large\frac{1}{(\log _ea)^3}$$.dt$
$\qquad= \large\frac{1}{(\log _ea)^3}. \frac{at}{\log_ea}$$+c \qquad= \large\frac{a^t}{\log_ea}$$+c$; again put values of t
=>$\large\frac{a^{\large a^{\Large a^{\Large a^{x}}}}}{(\log _e a)^4}+c$
Hence a is the correct answer.