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The power rule is a fundamental concept in calculus used to find the antiderivative of a function. It is particularly useful when dealing with functions that can be expressed as a power of a variable. For any function of the form \(x^n\), the power rule states that its antiderivative can be found using the formula:

• \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n eq -1\).

This rule allows us to reverse the process of differentiation by increasing the exponent by one and dividing by the new exponent.
To find the antiderivative of \(\sqrt{x}\), which can be rewritten as \(x^{1/2}\), we add one to the exponent \(1/2\), obtaining \(3/2\). Then, the expression becomes \(\frac{x^{3/2}}{3/2}\), which simplifies to \(\frac{2}{3}x^{3/2}\).
The simplicity of this rule makes it a handy tool for solving many basic integration problems, especially for polynomial expressions.

Integration is the process of finding the antiderivative of a function. It is essentially the opposite of differentiation. While differentiation breaks down a problem into its instantaneous rate of change, integration rebuilds the original function from this rate.
There are two types of integration: definite and indefinite. Indefinite integration, which we are dealing with here, focuses on finding a family of functions with a given derivative.
For our problem \(f'(x) = \sqrt{x}\), we use integration to find the original function \(f(x)\). Knowing that \(\sqrt{x} = x^{1/2}\), we apply the power rule to integrate.
Integrating \(x^{1/2}\) gives us \(\frac{2}{3}x^{3/2}\). This showcases integration as a powerful mathematical tool that 'inverts' differentiation, revealing the broader family of functions.

When we integrate a function, we incorporate a key component known as the constant of integration, denoted by \(C\). This constant represents all possible vertical shifts of the antiderivative on a graph, acknowledging that differentiation removes any constant value.
The essence of the constant of integration lies in the fact that many different original functions can have the same derivative. For example, \(f(x) = \frac{2}{3}x^{3/2} + C\) can represent an infinite number of functions depending on the value of \(C\).

• Without \(C\), the solution would miss capturing the entire possibility of original functions.
• Adding \(C\) ensures that the antiderivative solution is complete.

Including the constant of integration in your answers is crucial whenever you solve an indefinite integral. It acknowledges the complete set of solutions that exist and shows the versatility of integration in addressing real-world problems.