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The Fundamental Theorem of Calculus is a cornerstone in the field of calculus. It's essential for connecting differentiation and integration, two of the main operations in calculus. The theorem is made up of two parts.

• The first part states that if a function is integrable over an interval, then the integral defines a function whose derivative is the original function. This means integration can "reverse" differentiation.
• The second part, which is more frequently used, tells us how we can compute a definite integral if we know an antiderivative of the function. If \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), then the definite integral of \( f \) from \( a \) to \( b \) is \( F(b) - F(a) \).

This powerful theorem allows us to evaluate definite integrals without having to calculate the entire area under a curve by traditional means. By identifying an antiderivative, we can bypass complex calculations and still find accurate areas efficiently.

An antiderivative, often interchangeably called an indefinite integral, is a function whose derivative returns the original function. In essence, finding an antiderivative is the reverse of finding a derivative.
To capture this concept, consider derivatives that measure change, while antiderivatives retrieve the original function from this measure.
When evaluating definite integrals, we need the antiderivative to compute the result. This measurement is often expressed as

• If \( F'(x) = f(x) \), then \( F(x) \) is an antiderivative of \( f(x) \).
• The general form includes a constant \( C \), making it \( F(x) + C \) since differentiation removes constants.
• The antiderivative of \( (x+2)^{1/2} \) involves increasing the exponent by 1 and dividing by this new exponent.

Thus, from the exercise, the antiderivative of \( \sqrt{x+2} \), which is \((x+2)^{1/2}\), is \( \frac{2}{3} (x+2)^{3/2} + C \). This step is crucial as it enables us to apply the Fundamental Theorem of Calculus later on.

Integration by Substitution is a technique used to simplify the integration process. It's akin to the reverse of the chain rule used in differentiation. If an integral seems complicated at first glance, substitution can break it down into a simpler form.
The basic idea is to change the variable and the differential in a way that the integral becomes more manageable.
Unfortunately, not all integrals require this method. In the original exercise, a direct approach was possible without substitution, demonstrating the versatility of integration techniques.

• Start by identifying the substitution that relates complex parts of the integral. This usually means setting \( u = g(x) \), where \( g(x) \) forms a part of the original integral.
• Next, compute \( du \) by differentiating \( u \).
• Rewrite the integral in terms of \( u \) and \( du \) for ease of integration.
• Finally, once integrated, revert back to the original variable \( x \).

This method can simplify the evaluation of integrals that initially appear daunting, though in some exercises like the definite integrals one provided here, direct application of calculus principles suffices.