91Ó°ÊÓ

In calculus, the power rule is a fundamental tool for integrating or differentiating expressions. For integration, the power rule is used when you need to integrate expressions of the form \( x^n \). The general formula for integration using the power rule is:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]where \( n eq -1 \) and \( C \) is the constant of integration.To apply the power rule:- Identify the power \( n \) in the term \( x^n \).- Increase \( n \) by 1 to get \( n+1 \).- Divide \( x^{n+1} \) by \( n+1 \).- Don't forget to add \( C \), the constant of integration. The power rule makes integration straightforward, especially for polynomial functions. For instance, in the expression \( 2x^{-2/3} \), the power \( n \) is \(-2/3 \). Applying the power rule simplifies it into \( 6x^{1/3} + C \), quickly integrating the term by transforming negative exponents into positive ones. Remember, this rule does not apply when \( n = -1 \), which requires the natural logarithm method.

When integrating a function, we often encounter an arbitrary constant, known as the constant of integration, denoted by \( C \). It arises because differentiation of a constant results in zero, meaning any constant added to a function does not change its derivative.Understanding the constant of integration ensures that you correctly represent all potential antiderivatives of a function, providing the most general solution.Reasons for including \( C \) are:

• It accounts for the family of curves that solve the differential equation.
• Ensures correctness for indefinite integrals, where there isn't a specific boundary condition.

For example, when integrating the constant 9, the operation \( \int 9 \, dx \) yields \( 9x + C \). Without \( C \), the answer would ignore other possible solutions that differ by a constant value. Therefore, always include the constant \( C \), unless specific conditions, such as limits, are given that define the exact value of the constant.

Solving integration problems step-by-step involves a systematic breakdown of the problem, making it easier to handle and solve. This method helps clarify each part of the integral, supporting a comprehensive understanding.Let's look deeper into the original exercise:**Step 1 - Break Down the Expression:**The first step involves simplifying complex expressions into manageable parts. In our case, the integral \( \int (2x^{-2/3} + 9) \, dx \) is split into two simpler ones: \( \int 2x^{-2/3} \, dx \) and \( \int 9 \, dx \). This simplification allows separate application of integration rules.**Step 2 - Integration Using the Power Rule:**Applying the power rule focuses on one term at a time. For \( 2x^{-2/3} \), we adjust the power \(-2/3\) by adding 1 and then simplifying the result. This integration step provides the term \( 6x^{1/3} + C \).**Step 3 - Integrating the Constant:**Recognize that integrating a constant, like 9, involves multiplying the constant by the variable \( x \), resulting in \( 9x + C \).**Step 4 - Combine and Conclude:**Finally, add integrated terms together to form the complete integrated expression: \( 6x^{1/3} + 9x + C \). This outcome encompasses all potential solutions, thanks to the constant \( C \).A step-by-step approach not only provides clarity but also ensures each integration rule is applied correctly, leading to accurate and complete solutions.