Final answer:
Using partial fraction decomposition is a common approach to integrate functions like ((1)/(1-x+x²)), but it can become complex with non-real factors. For high precision results, a numerical integration method may be more appropriate.
Step-by-step explanation:
The trick to evaluate an integral of ((1)/(1-x+x²)) from 0 to ((1)/(10)) to 6 decimal places is likely to involve partial fraction decomposition, since this is a rational function and partial fractions are typically used to break these types of functions down into simpler fractions that are easier to integrate. However, before proceeding with any technique, it would be essential to ensure that the denominator doesn't factor into real linear factors, which it doesn't in this case. Therefore, the decomposition might include complex factors, which can be more complex and not typically a 'trick.'
Given the complexity of the integral and the requirement for six decimal places of precision, it may actually be more practical to use numerical integration techniques for this problem, such as Simpson's rule or a numerical integration function on a calculator or computer algebra system.