91Ó°ÊÓ

The power rule is fundamental in calculus, especially for integration and differentiation. It states that the integral of a power function, such as \( x^n \), is given by:\[ \int x^n \, dx = \frac{1}{n + 1} x^{n + 1} + C \]Here, \( n \) represents any real number except \( -1 \), since this would create a division by zero. The rule simplifies the process of finding antiderivatives by essentially reversing the differentiation power rule. To apply the power rule:- Increase the exponent by 1.- Divide by the new exponent.- Don’t forget the constant of integration, \( C \), which accounts for any constant term that could have been differentiated to zero.In our exercise, the power rule was applied to each term after expansion. For \( x^3 \), it becomes \( \frac{1}{4} x^4 \), and for \( 3x \), the process yields \( \frac{3}{2} x^2 \). This rule simplifies integration tasks considerably, turning complex processes into basic arithmetic.

Polynomial expansion is a necessary step in integration, allowing complex expressions to be broken down into simpler parts. By distributing the terms, you make it easier to apply rules like the power rule to each term individually.### Expansion ProcessWhen you encounter a function like \( x(x^2 + 3) \), it is initially multiplied as a polynomial. Expanding it results in:\[ x(x^2 + 3) = x^3 + 3x \]This step is especially crucial when dealing with integrands that involve products of polynomials. Each term must be dealt with separately to correctly apply the next steps of integration.Expanding polynomials means:- Distribute each term across others.- Simplify the result as much as possible.- Make sure each term is ready for further manipulation or integration.In our exercise, this expansion set the groundwork for applying the power rule to find the integral easily.

The constant of integration, denoted \( C \), is an essential part of indefinite integrals. Its inclusion is due to the nature of integration as an inverse operation of differentiation. When a function is differentiated, any constant term disappears, as the derivative of a constant is zero.### Importance of \( C \)\( C \) ensures that all possible antiderivatives are included in the integral solution. Without \( C \), you would miss any constant that would have existed in the original function. This concept highlights:- The arbitrary nature of the constant in antiderivatives.- Why every indefinite integral includes \( C \).- Its role in providing the most general solution to integration problems.For our exercise, after finding the antiderivative, we add \( C \) to indicate the complete set of functions whose derivative corresponds to our expanded polynomial. Adding \( C \) makes our solution \( \int (x^3 + 3x) \, dx = \frac{1}{4} x^4 + \frac{3}{2} x^2 + C \). It is a crucial finishing step to properly solve integration problems.