_ - 2 ^ 2 (x^2+ ( -x +x ) x ) d x=

Step-by-step solution

The integral is separated into even and odd components: I = _ - /2 ^ /2 x^2 , d x + _ - /2 ^ /2 ( -x +x ) x , d x Since x^2 is even, the first integral evaluates to twice the integral from 0 to /2 : 2 ₀^ /2 x^2 , d x = 2 [ x^3 3 ]₀^ /2 = ^3 12 The second integrand is odd, as ( +x -x ) x = - ( -x +x ) x , leading to cancellation over symmetric limits. Hence, the integral is zero. Combining the results yields I = ^3 12 + 0 = ^3 12 .