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The concept of an antiderivative is fundamental in the world of calculus. It is simply a function that reverses the process of differentiation. In other words, if you have a function, taking its derivative gives you another function, and the antiderivative of this new function would take you back to the original one.

When tackling problems involving antiderivatives, it's important to know some basic rules. For example, the power rule, which says that the antiderivative of a function like \( x^p \) is \( \frac{x^{p+1}}{p+1} + C \), where \( C \) is the constant of integration. This constant represents an infinite number of possible antiderivatives, each differing by a constant.

However, for definite integrals, which have specified limits, this constant cancels out when evaluating from the upper to the lower limits. So, in our example, we look for an antiderivative of \( \frac{5}{(t+2)^{11}} \), rewriting the function as \( 5(t+2)^{-11} \) and then applying the power rule backwards to get \( \frac{5}{-10}(t+2)^{-10} \), without worrying about the constant \( C \).

The Fundamental Theorem of Calculus sits at the heart of integral calculus and creates a powerful connection between differentiation and integration. There are two parts to this theorem, but here we focus on the second part, which is directly related to evaluating definite integrals.

This part tells us that if we want to compute the area under a curve (which is what a definite integral does), we can simply take the antiderivative of the function (let's call it \( F \)) and evaluate it at the upper limit (\( b \)) and lower limit (\( a \)) of integration, and then take the difference, i.e., \( F(b) - F(a) \).

In the step-by-step solution provided, we first found the antiderivative of the given function and then applied the second part of the Fundamental Theorem of Calculus to evaluate the integral within the specified limits. By calculating \( F(b) - F(a) \), we obtain the exact area under the curve between those two points on the \( t \)-axis.

Integration limits define the particular segment of the curve or function that you're interested in. In the context of definite integrals, these limits are also known as the boundaries of integration and are usually noted as \( a \) (the lower limit) and \( b \) (the upper limit).

When you set the limits of an integral, \( \int_{a}^{b} \), you're telling the math world that you want to find the total area under the curve from point \( a \) to point \( b \). Changing these limits changes the result of the integral because you're essentially altering the section of the graph you're examining.

For the exercise given, the lower limit is \( -1 \) and the upper limit is 0. Thus we are looking at the area under the graph of the function \( \frac{5}{(t+2)^{11}} \) from \( t = -1 \) to \( t = 0 \). The correct application of these limits is crucial for the accurate computation of the integral as shown in the exercise solution.