Final answer:
To evaluate the given integral, we can use partial fraction decomposition. Factor the denominator, express the integrand as a sum of fractions, determine the constants, and integrate each fraction separately.
Step-by-step explanation:
To evaluate the integral ∫ (5x² + 2x - 5)/(x³ - x) dx, we can use partial fraction decomposition. First, factor the denominator x³ - x as x(x - 1)(x + 1).
Then, express the integrand as a sum of fractions with these factors as denominators: (5x² + 2x - 5)/(x³ - x) = A/x + B/(x - 1) + C/(x + 1), where A, B, and C are constants to be determined.
To find A, multiply both sides of the equation by x, and then set x = 0. By simplifying, we get 5A - 5 = 0, so A = 1.
Similarly, by setting x = 1 and x = -1, we can find the values of B and C. Once we have determined the values of A, B, and C, we can integrate each fraction separately and sum them up to find the value of the original integral.