₀^2 |x^2+2 x-3 | d x is equal to

Step-by-step solution

Let f(x) = x^2 + 2x - 3 . We factorize the quadratic expression as f(x) = (x+3)(x-1) . The roots of f(x) = 0 are x = -3 and x = 1 . In the interval [0, 2] , the function f(x) changes sign at x = 1 . For 0 x For 1 x 2 , x^2 + 2x - 3 0 , so |x^2 + 2x - 3| = x^2 + 2x - 3 . The integral becomes ₀^1 (-x^2 - 2x + 3) dx + ₁^2 (x^2 + 2x - 3) dx . Evaluating the first integral: [ - x^3 3 - x^2 + 3x ]₀^1 = - 1 3 - 1 + 3 = 2 - 1 3 = 5 3 . Evaluating the second integral: [ x^3 3 + x^2 - 3x ]₁^2 = ( 8 3 + 4 - 6 ) - ( 1 3 + 1 -