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Integration is a fundamental concept in calculus that allows us to determine the antiderivative of a function. One of the most common methods for finding antiderivatives is through the power rule for integration. This rule is similar to the power rule used in differentiation but applies in reverse. The power rule states that if you have an expression of the form \( x^n \), where \( n \) is not equal to -1, the integral of this expression is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Here's a simple way to remember it: - Increase the power by one (add 1 to the exponent).- Divide by the new power.Applying this rule to each term individually in a polynomial is straightforward, as seen in our exercise. We take each term from the integrand and apply this rule separately, providing a clear path to find the antiderivative of the overall polynomial. It's important to ensure that the power rule is correctly applied to each term to avoid errors. This rule is essential for dealing with polynomial expressions and significantly streamlines the process of integration.

When finding the antiderivative of a function, one critical component to always include is the constant of integration, denoted as \( C \). This constant signifies that there are infinitely many functions which are antiderivatives of the given function, each differing by a constant. Why is this important?When you differentiate a constant, the result is zero, making it disappear in the derivative. This means that when taking an antiderivative, we must account for any shifted versions of the original function that we might have differentiated from.In the context of our exercise, after computing the antiderivative of each term, the constant \( C \) is added to the final expression. It is written as:\[ 8x^{4} + 14x^{2} - 8.5x + C \]This" + C" ensures that we cover all possible original functions the integrand could have come from. Remember, adding the constant is a crucial last step in finding the antiderivative.

In integration, the term 'integrand' refers to the function you are integrating. The integrand can contain variables raised to powers, constants, or other familiar functions like trigonometric or exponential functions. It serves as the core part of the integral expression, dictated by what needs to be integrated.Taking a closer look at the integrand in our given exercise: \(32x^3 + 28x - 8.5\),- **32x^3** is a term where the power rule for integration will apply to find its antiderivative.- **28x** is another polynomial term that follows similar rules.- **-8.5** is a constant, which through integration becomes -8.5x.Understanding the integrand is crucial because it sets the stage for how you'll apply integration rules. Each part of the integrand must be carefully addressed to ensure each part of the final derived function is calculated accurately. As seen, correctly identifying and applying procedures for each term in the integrand results in correctly computing its antiderivative.