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In calculus, integration allows us to find an antiderivative, or a function whose derivative is the original function given. For the function \( f(x)=\sqrt{x}(2x-5) \), the integration process involves simplifying the expression, which makes it easier to handle.### Simplifying the Function One technique we often use in integration is simplifying the integrand to a form that's easier to integrate. In this example, simplifying involves distributing the square root and rewriting the function in terms of exponents:

• Original form: \( f(x) = \sqrt{x}(2x - 5) \)
• Simplified form with exponents: \( f(x) = 2x^{1.5} - 5x^{0.5} \)

### Integrating Each TermAfter simplification, the function is integrated term by term. Each term of the expression \( 2x^{1.5} - 5x^{0.5} \) can be handled separately:

• The antiderivative of \( 2x^{1.5} \) is \( \frac{4}{5}x^{2.5} \).
• The antiderivative of \( -5x^{0.5} \) is \( -\frac{10}{3}x^{1.5} \).

Once each term is integrated, their results are combined into a single expression, forming the most general antiderivative before considering any constants.

Understanding how to work with exponents and radicals is crucial when simplifying functions before integrating. In this exercise, we began with the function \( \sqrt{x}(2x - 5) \), which includes a radical. Here's how these concepts are applied:### Expressing Radicals as ExponentsRadicals such as square roots can be rewritten using exponents to make differentiation or integration easier.

• A square root \( \sqrt{x} \) can be rewritten as \( x^{0.5} \).

By expressing radicals as fractional exponents, we can use power rules more effectively during integration.### Combining and Simplifying Exponential TermsThe function \( \sqrt{x}(2x - 5) \) was distributed and rewritten as \( 2x^{1.5} - 5x^{0.5} \). Breaking it down:

• \( x^{1.5} = x^{1+0.5} = x\times x^{0.5} \)
• Translating radicals to exponents simplifies the process of applying integration rules, such as increasing the power by 1 and dividing by the new exponent.

Every antiderivative includes a constant of integration, denoted as \( C \). It accounts for the family of all possible antiderivatives of a function. The reason this constant is necessary is rooted in the fact that differentiation removes constants (as the derivative of a constant is zero).### Understanding \( C \) in AntiderivativesAfter finding the antiderivative for each term, we express the most general antiderivative as:\[ g(x) = \frac{4}{5}x^{2.5} - \frac{10}{3}x^{1.5} + C \]

• Here, \( C \) can be any real number, meaning the antiderivative represents a whole family of functions.
• This concept can be visualized as parallel curves on a graph, each shifted vertically depending on the value of \( C \).

The constant of integration is crucial in solving real-world applications, such as finding a specific solution to a physical problem when initial conditions are known. When integrating, remember to include \( C \) to ensure the solution's completeness.