The value of the integral _ - 3 ^ 3 (2x^9 + 3x^8 - 5x^7 + 9x^6 + 4x^3 - x + 3) x^2 + 3 , dx is

Step-by-step solution

Let the given integral be I . We can split the integrand into odd and even functions: I = _ - 3 ^ 3 2x^9 - 5x^7 + 4x^3 - x x^2 + 3 , dx + _ - 3 ^ 3 3x^8 + 9x^6 + 3 x^2 + 3 , dx The first integral is 0 because the integrand is an odd function and the limits are symmetric about the origin. For the second integral, simplifying the integrand gives: I = _ - 3 ^ 3 3x^6(x^2 + 3) + 3 x^2 + 3 , dx = _ - 3 ^ 3 ( 3x^6 + 3 x^2 + 3 ) , dx Since the integrand is an even function, we can write: I = 2 ₀^ 3 ( 3x^6 + 3 x^2 + 3 ) , d