_ - / 2 ^ / 2 ^2 x d x is equal to

Step-by-step solution

Let I = _ - / 2 ^ / 2 ^2 x dx . Since f(x) = ^2 x is an even function, f(-x) = ^2(-x) = (- x)^2 = ^2 x = f(x) . Therefore, I = 2 ₀^ / 2 ^2 x dx . Using the identity ^2 x = 1 - (2x) 2 , we have I = 2 ₀^ / 2 1 - (2x) 2 dx . I = ₀^ / 2 (1 - (2x)) dx . I = [x - (2x) 2 ]₀^ / 2 . I = ( 2 - ( ) 2 ) - (0 - (0) 2 ) . I = 2 - 0 - 0 + 0 = 2 . Answer: 2