If .I_n= _ / 2 ^ e^ -x ^n x d x ] , then I₂₀₁₈ I₂₀₁₆ =

Step-by-step solution

We have, I_n= _ / 2 ^ e^ -x ^n x d x Integration by parts gathered I_n= [-e^ -x ^n x ]_ / 2 ^ - _ / 2 ^ -e^ -x n ^ n-1 x(- x) d x =0- _ / 2 ^ n e^ -x ^ n-1 x x d x gathered By integration parts again, we get gathered =-n [ (-e^ -x ^ n-1 x x ]_ / 2 ^ . .- _ / 2 ^ -e^ -x ( ^n x-(n-1) ^ n-2 x ^2 x ) d x ] =-n [0+ _ / 2 ^ (e^ -x ^n x-(n-1) e^ -x . . . ^ n-2 x (1- ^2 x ) d x ] [ ^2 x=1- ^2 x ] gathered array cc =-n [ _ / 2 ^ e^ -x ^n x d x-(n-1) . & . _ / 2 ^ e^ -x ^ n-2 x d x+(n-1) _ / 2 ^ e^ -x ^n x d x ] & =-n [I_n-(