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The Power Rule of Integration is a fundamental technique in calculus used for finding the antiderivative or indefinite integral of powers of x. It states that for any real number n, which is not equal to -1, the integral of x^n with respect to x is (x^(n+1))/(n+1), plus a constant of integration, C. In simpler terms, to integrate a power of x, increase the exponent by one and then divide by this new exponent.

For example, if you're asked to integrate x^(1/2), according to the Power Rule, you would add 1 to the 1/2 to get 3/2 and then divide by 3/2. In doing so, the integral becomes (2/3)x^(3/2). This approach is used to integrate each term in a polynomial or a sum of monomials, as seen in the exercise provided. When each term is integrated this way, it simplifies the process of finding the overall antiderivative for the function.

An Antiderivative, also known as an indefinite integral, is essentially the reverse of taking a derivative. While a derivative represents the rate of change of a function, an antiderivative is a function whose rate of change (derivative) is the original function that you started with. Finding an antiderivative involves integrating the function without the limits of integration that you would find in a definite integral.

For the function x^(1/2) + x^(1/4), we would find the antiderivative by integrating each term separately. As shown in the solution, the antiderivative of x^(1/2) is (2/3)x^(3/2) and the antiderivative of x^(1/4) is (4/5)x^(5/4). Remember that since these are indefinite integrals, a constant C would usually be added at the end, but in the context of a definite integral, this constant cancels out.

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration, showing that they are inverse processes. It consists of two parts. The first part states that if a function is continuous on a closed interval [a, b], then the function has an antiderivative on that interval, and the integral of the function over [a, b] can be computed using any one of its antiderivatives.

The second part of the theorem is directly used to solve definite integrals. It states that if F(x) is the antiderivative of a continuous function f(x), then the definite integral of f(x) from a to b is given by F(b) - F(a). This allows us to simply evaluate the antiderivative at the bounds of integration, subtracting the lower bound from the upper bound, providing the net area under the curve from a to b. In the given exercise, this method simplifies the calculation from integrating a function over an interval, to simply evaluating a function at two points and subtracting those values.