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Integrals are a fundamental concept in calculus, representing the area under a curve on a graph. They help in finding quantities such as total accumulation of change over time. Imagine drawing a curve on a graph representing the speed of a car over time; the integral of this graph would give you the total distance traveled. This process of finding an integral is known as integration.

There are two main types of integrals, definite and indefinite. The integral we are dealing with in this problem is a definite integral. Definite integrals are used when we are interested in the actual numerical value of the area under the curve between two points. These two points are called the limits of integration, and they define the range over which you calculate the integral.

In the problem, we have the integral from 0 to 2 of a specific function, \[\int_{0}^{2} \frac{d x}{(x-1)^{2}}\]. This tells us to calculate the accumulation of the function's values between these two limits.

Antiderivatives, also known as indefinite integrals, are functions that reverse the process of differentiation. They help us understand the original function from its rate of change or derivative. Essentially, if you know the rate at which something is increasing, the antiderivative tells you the overall change over time.

Finding an antiderivative involves applying a set of rules, one of which is the power rule. It states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(n\) is a constant not equal to -1. In our exercise, the function \(\frac{1}{(x-1)^2}\) was converted to its antiderivative using the power rule as \(-\frac{1}{x-1} + C\).

The constant \(C\) is used here because antiderivatives are not unique. There can be infinitely many such functions, each varying by a constant. When calculating a definite integral, however, this constant cancels out when evaluating the integral at the upper and lower limits.

The divergence of integrals is a concept that arises when the area under a curve becomes infinite. This usually happens when there's a point within the range of integration where the function under the integral becomes undefined, such as dividing by zero. It is crucial to identify if a given integral converges to a finite value or diverges to infinity.

In the exercise provided, the function \(\frac{1}{(x-1)^{2}}\) has a point of concern at \(x = 1\). At this point, the denominator becomes zero, which could potentially make the integral diverge. However, in our specific bounds from 0 to 2, when computed, the integral results in a finite value \(-2\).

Thus, it shows the importance of carefully evaluating the limits and checking if the points causing divergence lie within or outside the integration bounds. This helps in deciding whether the result is finite or infinite.