Final answer:
To rewrite the rational expression using partial fractions, factor the denominator and express the expression as the sum of two fractions. To find the indefinite integral, integrate each term individually using the partial fractions decomposition.
Step-by-step explanation:
To rewrite the rational expression using the method of partial fractions, we start by factoring the denominator: x³ - x + 3 = (x - 1)*(x² + x + 3). The factors of the denominator are x - 1, x² + x + 3. Next, we express the rational expression as the sum of two fractions: (x² + x - 2)/(x³ - x + 3) = A/(x - 1) + (Bx + C)/(x² + x + 3). We then find the values of A, B, and C by equating the numerators on both sides.
Now, to find the indefinite integral of x² + x - 2 / x³ - x + 3, we can use the partial fractions decomposition. After decomposing the rational expression into A/(x - 1) + (Bx + C)/(x² + x + 3), we can integrate each term individually. The integral of A/(x - 1) can be found by using the substitution method, and the integral of (Bx + C)/(x² + x + 3) can be found by using the method of partial fractions.
The final answer to the indefinite integral of x² + x - 2 / x³ - x + 3 after performing the integrations is (A * ln|x - 1|) + (B * ln|x² + x + 3|) + (C * atan((2x + 1) / sqrt(8))) + C, where A, B, and C are the constants obtained from the partial fractions decomposition.