The value of _ - 2 ^ 2 ^2 x 1+2^x ~d x is

Step-by-step solution

Let I = _ - 2 ^ 2 ^2 x 1+2^x ~d x ...(i) I = _ - 2 ^ 2 ^2 x 1+2^ -x ~d x ...(ii) [ _ a ^ b f (x) d x= _ a ^ b f ( a + b -x) d x ] Adding (i) and (ii), we get aligned & 2 I = _ - 2 ^ 2 ^2 x ~d x & 2 I =2 ^ 2 ^2 x ~d x aligned [ array l _ -a ^a f(x) d x=2 ₀^a f (x) d x if f (x) is an even function array ] aligned I & = ₀^ 2 ( 1- 2 x 2 ) d x & = 1 2 [x- 2 x 2 ]₀^ 2 & = 1 2 ( 2 -0 )= 4 aligned