If _ x 0 ( _ r=1 ^n r 2 n ) ( _ r=1 ^n ^2 r 2 n ) ( _ r=1 ^n ^3 r 2 n ) ( _ r=1 ^n ^4 r 2 n ) (e^ 1 n -1 )^4= k ^2 then find the value of 1 k .

Step-by-step solution

aligned & ₀^1 ( 2 x ) d x ₀^1 ^2 ( 2 x ) d x ₀^1 ^3 ( 2 x ) d x ₀^1 ^4 ( 2 x ) d x & = ( 2 )^4 ( ₀^ / 2 ^2 t d t ) ( ₀^ / 2 ^3 t d t ) ( ₀^ / 2 ^4 t d t )= ( 2 )^4 1 4 2 3 3 16 = 1 2 ^2 = k ^2 & 1 k =2 & aligned