Final answer:
The integral of (2x² - 9x - 9) / (x³ - 9x) dx requires simplification and appropriate integration techniques such as polynomial long division, partial fractions, substitution, or integration by parts for evaluation.
Step-by-step explanation:
The integral given to evaluate is ∠ 2x² - 9x - 9 / x³ - 9x dx. This integral seems complicated due to the polynomial expressions both in the numerator and the denominator, however, the solution typically involves simplifying the expression and then using methods like polynomial long division, partial fractions, substitution, or integration by parts to solve it.
Firstly, it's often prudent to simplify the fraction if possible. If the numerator can be factored to have a common factor with the denominator, that would significantly simplify the integral. Next, if simplification isn't directly possible, polynomial long division or partial fraction decomposition could be applied, breaking the integral into simpler, more manageable parts.
Integration techniques such as substitution might come into play if the integral is broken down into simpler fractions. In some cases, if the derivative of the denominator is present in the numerator, integration by parts might be a helpful strategy although it's more common in integrals involving products of functions of different types (e.g., polynomials and exponentials).
Due to the complexity of the integral, a detailed step-by-step solution would require more context or simplification to ensure an accurate answer.