91Ó°ÊÓ

When we talk about finding an antiderivative, we're discussing a key concept in calculus. The process involves finding a function that, when differentiated, gives back the original function. In this exercise, the original function is \( 2x^3 - 5x + 7 \). The goal is to reverse the differentiation process.
To clarify, let's think of differentiation as "doing." If we differentiate a function, we are "doing" calculus to it. Finding an antiderivative is like "undoing" this operation. It's akin to taking a backwards step in the calculus world.
Antiderivatives are not unique; they differ by a constant. This is because when differentiating a constant, you get zero, so it can "hide" when you differentiate. This is why we often add a constant \( C \) when expressing an indefinite integral. It accounts for all possible original constants that wouldn't appear in the derivative.

The power rule for integration is a straightforward technique used to find the antiderivatives of functions involving powers of \( x \). For any power \( n \), the rule is: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
This rule is immensely useful because it allows us to quickly integrate terms like \( 2x^3 \) or \( -5x \). By following the power rule, we smoothly transition from derivative to antiderivative. Let's see how it works:

• For \( 2x^3 \): Apply the rule to get \( \frac{2x^{4}}{4} = \frac{x^4}{2} \).
• For \( -5x \): Recognize this as \( -5x^1 \) and apply the rule to get \( -\frac{5x^2}{2} \).
• For a constant, like \( 7 \), simply multiply by \( x \) as the "power" is zero, resulting in \( 7x \).

By applying this rule separately to each term, we craft the antiderivative for the entire function.

The constant of integration, often denoted as \( C \), is an essential part of finding an indefinite integral. When we integrate, we're essentially going backwards from a derivative to find the "original" function. However, differentiation has a property where any constant becomes zero.
This means when going back, we can end up "missing" any constant term that was originally there as it does not show up in the derivative. Thus, \( C \) accounts for all these possibilities.
We always add \( C \) because:

• The result is a family of functions, not a single one.
• It signifies that there are infinitely many functions that could have the same derivative.
• It ensures the result is as general as possible, encompassing all potential original functions.

For our solution, \( C \) ensures that \( \frac{x^4}{2} - \frac{5x^2}{2} + 7x + C \) represents every potential antiderivative of the given function.