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In calculus, the term antiderivative refers to a function whose derivative is the given function. For instance, when dealing with the function f(t) = 4t + 1, finding the antiderivative means looking for a function F(t) such that F'(t) = 4t + 1. The process of finding antiderivatives is known as integration. Each antiderivative is one of infinitely many functions that differ only by a constant, C, and that set of functions is called the indefinite integral of f(t).

In our example, the antiderivative of 4t + 1 is 2t^2 + t + C. Understanding the antiderivative is crucial in solving integration problems, especially when applying the Fundamental Theorem of Calculus, which enables us to evaluate definite integrals.

A definite integral has both a lower and upper limit, and it represents the net area under the curve of the function between those limits. In contrast to an indefinite integral, a definite integral has a numerical value and does not include the constant of integration, C. To find a definite integral, as in the exercise \(F(x) = \int_{x}^{x+2}(4t+1) dt\), we evaluate the antiderivative at the upper limit and subtract its evaluation at the lower limit.

The Fundamental Theorem of Calculus parts 1 and 2 connect differentiation and integration, showing that an antiderivative can be used to compute a definite integral. In the given problem, the subtraction of the evaluations of the antiderivative at x + 2 and x simplifies the process and gives us the exact net area between x and x + 2 under the curve described by the function 4t+1.

The process of differentiation involves finding the derivative of a function, which is the rate at which the function is changing at any given point. For our function F(x), the differentiation step involves applying the derivative operator d/dx to the function. According to the Fundamental Theorem of Calculus, the derivative of an integral of a function brings us back to the original function, given some conditions are met.

In the context of the practice problem, taking the derivative of F(x) after calculating the definite integral gives us F'(x) = 8x + 9. This simplified expression represents the slope of the tangent line to the curve of F(x) at any point x, and in turn, it describes the instantaneous rate of change of the area under the curve from x to x+2 as x changes.