Write the following function as the sum of an odd and an even function. Then calculate the integral over the given interval using the properties of odd and even functions.

Question

(x-1)^3, I=[-1,1]

Answer 1

x(x^2-2x+1)
-1(x^2-2x+1)

= x^3 - 2x^2 + x - x^2 + 2x - 1

= (x^3 + 3x)odd ( -3x^2 -1) even

the odd part will give zero (+ on one side of x = 0 and - on the other)
so
-x^3 - x = - (x^3+x)so (x^3+x) at -1 - at +1
-1-1 - 1-1 = -4