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The Power Rule is a fundamental principle when working with integrals, crucial for calculating antiderivatives easily. This rule states that when you have a function of the form \(x^n\), its antiderivative can be found using the formula:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]where \(n\) is any real number except \(-1\), and \(C\) is the constant of integration. The Power Rule makes integrating expressions straightforward once you've rewritten them to match the \(x^n\) form.
In our problem, the term \(4x^{-3/2}\) becomes: \[4 \cdot \frac{x^{(-3/2) + 1}}{(-3/2) + 1}\]Calculating within the fraction, you end up with:\[\frac{x^{-1/2}}{-1/2},\]further simplifying gives us:\[-8x^{-1/2} = -8\sqrt{x}\]Apply the Power Rule consistently, step-by-step, to each component when integrating. It's a powerful shortcut, streamlining the path to the solution.

An indefinite integral is essentially the antiderivative of a function. It represents a family of functions and is often expressed with an integrand and a constant of integration, \(C\). When calculating indefinite integrals, you're finding a general form of antiderivatives without set bounds.To compute the indefinite integral of \(f(x)\), you integrate all parts of \(f(x)\) separately:\[\int(4x^{-3/2} + 6x^{-5/2}) \text{d}x\]Each term is approached individually:

• \(4 \cdot \int x^{-3/2}\text{d}x\)
• \(6 \cdot \int x^{-5/2}\text{d}x\)

This process helps ensure clarity and structure in your computations. Indefinite integrals allow flexibility in exploring potential solutions, given the constant \(C\) which adjusts based on any additional conditions provided, like \(F(1) = 4\).As you integrate, you're reversing differentiation, returning to the function that, when derived, gives the original \(f(x)\). The indefinite integral is thus foundational in calculus.

The Constant of Integration, denoted as \(C\), arises whenever you deal with indefinite integrals. Since differentiation erases constant terms, integrating does not recover them; \(C\) accounts for any such omissions. It's what transforms the indefinite integral from a single path back to a family of functions.When tasked with finding a specific antiderivative, extra information is crucial. Given \(F(1)=4\), this condition lets you pinpoint \(C\) precisely for your function:To do this, substitute the given \(x\) value into the expression already obtained from the indefinite integral:\[F(x) = \frac{-8\sqrt{x} - 4}{\sqrt{x^3}} + C\]Setting \(x = 1\), you need:\[4 = \frac{-8\cdot\sqrt{1} - 4}{\sqrt{1^3}} + C\]Solving this equation, you find \(C = 16\). This specific \(C\) ties the particular solution directly to the provided condition.In encompassing the entire spectrum of possible integral outcomes, the constant of integration plays a pivotal role, ensuring that all bases are covered in indefinite integrals.