( ⁻¹ [ _ - / 2 ^ / 2 x 1+e^x d x ]= )

Step-by-step solution

Given, ( ⁻¹ [ _ - 2 ^ 2 x 1+e^x d x ] ) Let (I= _ - 2 ^ 2 x 1+e^x d x )...(i) ( aligned & I= _ - 2 ^ 2 ( 2 - 2 -x ) .1+e^ ( 2 - 2 -x . ) d x & [ _a^b f(x) d x= _a^b f(a+b-x) d x ] & = _ - 2 ^ 2 x 1+e^ -x d x= _ - 2 ^ 2 e^x x 1+e^x d x (ii) aligned ) Adding Eqs. (i) and (ii), we get ( gathered 2 I= _ - 2 ^ 2 (e^x+1 ) x (1+e^x ) d x (iii) = _ - 2 ^ 2 x d x=[ x]_ - 2 ^ 2 2 I=[1+1]=2 I=1 ⁻¹ [ _ - 2 ^ 2 x 1+e^x d x ]= ⁻¹(1)= ⁻¹ ( 4 )= 4 gathered )