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The concept of the antiderivative is essential in calculus. It involves reversing differentiation, which gives us the original function from its derivative. This is also known as finding the indefinite integral of a function. The notation used for this process is the integral sign \( \int \), followed by the function and the differential \( dx \). This notation indicates that you are finding the antiderivative of a function with respect to \( x \).
For instance, if you start with an expression like \( \int \left( \frac{\sqrt{x}}{2} + \frac{2}{\sqrt{x}} \right) dx \), your task is to determine the function \( F(x) \) such that \( F'(x) \) equals the given expression. This process is important because determining antiderivatives enables you to solve real-world problems involving areas under curves and total accumulated quantities.
Remember, when finding an antiderivative, it is crucial to include a constant of integration, \( C \). This represents the infinite number of vertical shifts that all have the same derivative and thus constitute the same general antiderivative.

The power rule is a fundamental technique used to find antiderivatives of functions written as powers of \( x \). This rule states that:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]where \( n eq -1 \). This rule applies to each term separately when dealing with a sum of terms, making complex problems more manageable.
For example, in the given problem, you simplify \( \frac{\sqrt{x}}{2} + \frac{2}{\sqrt{x}} \) into the exponent form as \( \frac{1}{2}x^{1/2} + 2x^{-1/2} \) to apply the power rule individually. Integrating each term:

• For \( \frac{1}{2}x^{1/2} \), apply the rule to find \( \int \frac{1}{2}x^{1/2} \, dx = \frac{1}{3}x^{3/2} \).
• For \( 2x^{-1/2} \), apply the rule to find \( \int 2x^{-1/2} \, dx = 4x^{1/2} \).

By applying the power rule correctly, you calculate the individual antiderivatives, which are then summed up with a constant, \( C \), to give the general antiderivative of the expression.

The differentiation check is a crucial step when finding antiderivatives. After you determine an antiderivative, differentiating it should yield the original function or expression.
Why is this important? It verifies that you have performed the integration correctly and arrived at the right antiderivative.
Take the antiderivative we found earlier: \( \frac{1}{3}x^{3/2} + 4x^{1/2} + C \). Upon differentiating each term, you should arrive back at the initial terms:

• Differentiating \( \frac{1}{3}x^{3/2} \), you have: \( \frac{d}{dx} \left( \frac{1}{3}x^{3/2} \right) = \frac{1}{2}x^{1/2} \).
• Differentiating \( 4x^{1/2} \), you find: \( \frac{d}{dx} \left( 4x^{1/2} \right) = 2x^{-1/2} \).

Adding these differentiated values together should produce the expression inside the integral: \( \frac{1}{2}x^{1/2} + 2x^{-1/2} \), confirming your integration was executed correctly.