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The Power Rule is a fundamental tool in calculus for finding the antiderivative of a function. It's especially useful for integrating polynomial functions or functions written in terms of powers of a variable.
When we say the Power Rule, we are referring to the formula: \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]This rule allows you to find the antiderivative of \(x^n\), where \(n\) is a constant.
The key point to remember is:

• Add 1 to the exponent \(n\) of the variable \(x\).
• Divide by the new exponent \(n+1\).

This process gives us the antiderivative of the function.
Applying this to a specific example, when you have a function like \(x^{-1/2}\), using the Power Rule involves:

• Adding 1 to \(-1/2\), which results in \(1/2\).
• Dividing the base raised to \(1/2\) by \(1/2\), which is \(2x^{1/2}\).

Thus, the antiderivative of \(x^{-1/2}\) is \(2x^{1/2}\).

An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function you started with. It essentially reverses the process of differentiation.
For the function \(f(x) = x^n\), one possible antiderivative is \(F(x) = \frac{x^{n+1}}{n+1}\).
Antiderivatives are not unique, they always come with a constant \(C\), because when you differentiate a constant it becomes zero. That's why you add \(+C\) to the result:

• \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]
• More than one antiderivative can exist for a function.
• This constant accounts for the constant term in the original function that was "lost" when the derivative was taken since the derivative of a constant is zero.

In the given exercise, the integration of \(x^{-1/2}\) gives the antiderivative \(2x^{1/2}\), confirmed by the fact that the derivative of \(2x^{1/2}\) is \(x^{-1/2}\).

When working with definite integrals, limits of integration play a crucial role, as they define the interval over which the integration is performed.
A definite integral consists of an antiderivative accompanied by limits of integration, depicted as the numbers at the bottom and top of the integral sign:\[\int_{a}^{b} f(x) \, dx\]Here, \(a\) and \(b\) are the lower and upper limits of integration, respectively.

• The result of evaluating a definite integral \, includes using the Fundamental Theorem of Calculus, which connects differentiation with integration.
• This theorem states that if \(F(x)\) is an antiderivative of \(f(x)\), then:\[\int_{a}^{b} f(x) \, dx = F(b) - F(a).\]

By substituting the limits of integration into the antiderivative, you find the net area under the curve \(f(x)\) from \(x=a\) to \(x=b\).
In practical terms, after you have determined the antiderivative, compute it at both limits of integration, and subtract the two results.
As shown in the solution, substituting into \(2x^{1/2}\) allows calculation from \(x=1\) to \(x=4\), yielding the solution \(2\sqrt{5}\). This demonstrates how the definite integral is the specific net value obtained between the chosen limits.