_ -2 ^2 |x^2-x-2 | dx =

Step-by-step solution

The definite integral _ -2 ²|x^2 - x - 2|dx requires analyzing the absolute value function. The quadratic x^2 - x - 2 = (x-2)(x+1) has roots at x = -1 and x = 2 , and is negative between these roots while positive elsewhere. The integral splits into two parts: _ -2 ⁻¹(x^2 - x - 2)dx + _ -1 ²(-x^2 + x + 2)dx Evaluating the first integral: [ x^3 3 - x^2 2 - 2x ]_ -2 ⁻¹ = 11 6 Evaluating the second integral: [- x^3 3 + x^2 2 + 2x ]_ -1 ² = 9 2 The total integral is 11 6 + 9 2 = 19 3 . Final answer: 19 3