What is the value of _ -1 ² 1 - x, 1 - x^3 , dx ?

Step-by-step solution

Let I = _ -1 ² 1 - x, 1 - x^3 , dx . We can rewrite the integrand using the property a - b, a - c = a - b, c . Thus, 1 - x, 1 - x^3 = 1 - x, x^3 . The integral becomes: I = _ -1 ² (1 - x, x^3 ) , dx = _ -1 ² 1 , dx - _ -1 ² x, x^3 , dx The first integral is simply: _ -1 ² 1 , dx = [x]_ -1 ² = 2 - (-1) = 3 To evaluate the second integral, we determine the intervals where x x^3 and x^3 x by analyzing the sign of x - x^3 = x(1 - x)(1 + x) . The roots are -1, 0, 1 . In [-1, 0] , x^3 x x, x^3 = x^3 In [0, 1] , x x^3 x,