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

Step-by-step solution

Let I = _ -1 ^1 ( 1+x+x^2 - 1-x+x^2 ) dx . The integrand f(x) = 1+x+x^2 - 1-x+x^2 satisfies f(-x) = -f(x) , making it an odd function. Since the integration interval [-1, 1] is symmetric about the origin, the integral of any odd function over such an interval is zero. Therefore, I = 0 .