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Integration is the process of finding the antiderivative or the indefinite integral of a function. It is essentially the reverse operation of differentiation. Whereas differentiation gives us the rate of change or the slope of a function, integration gives us the accumulated total or area under the curve of a function. For instance, if you have a function describing velocity, integrating it can help you find the displacement. In the context of the exercise, the task is about finding the antiderivative, meaning to determine which function, when differentiated, results in the given function. Here's a step-by-step approach to basic integration:

• Identify the function you wish to integrate.
• Look for patterns that match integration rules, like the power rule.
• Apply the relevant integration formula.
• Always add a constant of integration, denoted as C, because integration can have multiple solutions.

The power rule for integration is one of the simplest and most commonly used techniques for finding antiderivatives. It is applicable to functions of the form \(x^n\) where \(neq -1\). The rule is expressed as:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]where C is the constant of integration, which appears because the derivative of a constant is zero, leaving flexibility in what the original function could have been.To use the power rule:

• Increase the exponent by 1.
• Divide by the new exponent.
• Don’t forget to include the constant C at the end.

Example: Let’s say we are to integrate \(x^{-2}\). According to the power rule, \[ \int x^{-2} \, dx = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C \] Make sure to adjust the power and coefficient accordingly, ensuring everything is neatly simplified.

Differentiation is the process of finding the derivative of a function. The derivative represents the rate of change of a function or how it changes over time. It’s a fundamental tool in calculus that helps us understand how a function behaves, finding slopes, rates of growth, and approximations.In checking the correctness of an antiderivative, differentiation is used. You take your calculated antiderivative and find its derivative. If the derivative matches the original function you started with, your antiderivative is correct.For example, in the exercise:

• The antiderivative of \( \frac{1}{x^2} \) was found to be \( -\frac{1}{x} + C \).
• By differentiating \( -\frac{1}{x} + C \), we go through the derivative process to get back to \( \frac{1}{x^2} \).
• If verified correctly, the calculated antiderivative reflects the original function.

The purpose here is to double-check your work, ensuring no integration steps were skipped or miscalculated.

Verification ensures that the solution to a problem is correct and valid. After finding an antiderivative, it’s crucial to verify by differentiating that solution and checking if it reverts to the original function. This acts as a confirmation step to avoid any slips during calculations.Double-checking involves:

• Taking the derivative of the antiderivative.
• Comparing the resulting function with the original given function.
• If both match, the antiderivative is confirmed correct.

For the exercise's antiderivatives, each was verified separately. For instance, upon obtaining \(2x - \frac{5}{x} + C\), differentiating it led to the original function \(2 - \frac{5}{x^2} \), thus verifying correctness.Verification is an excellent habit in mathematics to ensure solutions are reliable and that all integrations have been correctly executed.