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Integration is a fundamental technique used in calculus to calculate the area under curves, among other applications. One effective method for integrating certain types of functions is the partial fraction decomposition, particularly for rational functions. This approach breaks down a complex rational function into simpler fractions that are easier to integrate individually.

In our given problem, the function \( \frac{1}{(x-1)^2(x-2)} \) needs to be integrated. Using partial fraction decomposition helps to simplify this function by expressing it as a sum of simpler fractions.

Once the function is decomposed correctly, each of the resulting simpler fractions can be integrated using basic integration techniques. For linear factors, this often involves logarithmic integration or simple power rules, while for repeated factors, more attention needs to be paid to the order of integration.

Linear factors are expressions of the form \( ax + b \), representing straight line equations. In partial fraction decomposition, knowing how to decompose a rational function with linear factors is crucial for correct integration. In the example function \( \frac{1}{(x-1)^2(x-2)} \), \( (x-1) \) and \( (x-2) \) are linear factors.

When one of these linear factors is repeated, such as \( (x-1)^2 \), both \( \frac{1}{x-1} \) and \( \frac{1}{(x-1)^2} \) should be included in the decomposition. Each factor accounts for a part of the original function that needs simplification. Failing to incorporate a term for a repeated factor, as initially shown in the exercise, results in an incomplete decomposition, leading to errors in the integration process. It is essential to handle linear and repeated factors correctly for a valid decomposition.

Rational functions are quotients of polynomial functions. They take the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. Using partial fraction decomposition on rational functions simplifies the integration process.

For correct decomposition, the degree of the numerator should be less than the degree of the denominator. If not, polynomial long division is necessary before decomposing. In the problem example, the degree of \( (x-1)^2(x-2) \) is sufficient for partial fraction decomposition as it stands because it is greater than the degree of the numerator.

This process ultimately supports the integration efforts by allowing mathematicians to focus on simpler and more manageable expressions which unravel the complexity of the original rational expression.