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The Fundamental Theorem of Calculus (FTC) plays a crucial role in bridging the gap between differentiation and integration. It's like a bridge that connects two major concepts of calculus. The FTC consists of two parts.

• First, it establishes the relationship between the derivative and the integral of a function.
• Second, it provides a practical technique for evaluating definite integrals.

According to the theorem, if you have a function that is continuous over a closed interval \([a, b] \), and if \(F(x)\) is an antiderivative of \(f(x)\) on that interval, then:\[F(b) - F(a) = \int_a^b f(x) \, dx\]This means that you can find the total accumulation of values of \(f(x)\) over \([a, b]\) by simply evaluating its antiderivative at the endpoints of the interval. In the given problem, differentiating the integrals using FTC simplifies the equations and allows us to express them in a form that can be solved directly. This simplification aids in finding resulting functions or values, such as \(f(2)\) in the example.

Continuous functions are essential in calculus because they ensure certain predictable behaviors without sudden jumps or breaks. Consider a continuous function \(f(x)\) over a range — it means for any two points, no breaks occur in the path drawn by \(f(x)\). This continuity is important for applying integrations and differentiations smoothly.In calculus, especially when using the FTC, continuity ensures that the antiderivative exists over the interval of interest. Thus, we can always differentiate back appropriately. For example, in the exercise, we start with a continuous function \(f:[0, \infty) \rightarrow \mathbb{R}\). This guarantees us that each integral from \(0\) to any given point behaves nicely.Continuity also means that small changes to the value of \(x\) will result in small changes in \(f(x)\), making predictions possible and mathematical manipulation convenient. This property is what allows step-by-step solving using calculus principles with confidence.

Integration techniques are numerous, each serving to solve different kinds of problems. In calculus, integration is the process of finding an antiderivative or the area under a curve. For solving the problem statement effectively, you need familiarity with various techniques and understand when to apply each. In the exercise, the primary method involves using the FTC, yet other integration techniques like substitution or partial fractions can be critical in other scenarios. Here are some typical techniques:

• Substitution: Useful to simplify expressions before integrating.
• Integration by parts: A way to integrate products of functions.
• Partial fraction decomposition: Breaks down complex rational expressions.

In the exercise, all integrals were attacked by directly differentiating, rooted in the FTC. This approach revealed new forms of equations where functions like \(f(x)\) could be easily expressed in practical terms. The understanding of such techniques empowers you to tackle diverse calculus problems efficiently, and knowing when and how to apply them is key to simplifying complex integrals.