Problem 25 - Applied Calculus

Question

Find an antiderivative. $$f(x)=x+x^{5}+x^{-5}$$

Step-by-Step Solution

Answer

\( F(x) = \frac{x^2}{2} + \frac{x^6}{6} - \frac{x^{-4}}{4} + C \)

Step 1: Understand the Antiderivative

The antiderivative, also known as the indefinite integral, is a function whose derivative is the given function. The goal is to find a function, say \( F(x) \), such that \( F'(x) = f(x) \).

Step 2: Apply the Power Rule for Antiderivatives

To find the antiderivative, apply the power rule. The power rule for antiderivatives states that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration, and \( n eq -1 \).

Step 3: Find the Antiderivative of Each Term

Apply the power rule to each term of the function:1. For \( x \), \( n = 1 \), so the antiderivative is \( \frac{x^{1+1}}{1+1} = \frac{x^2}{2} \).2. For \( x^5 \), \( n = 5 \), so the antiderivative is \( \frac{x^{5+1}}{5+1} = \frac{x^6}{6} \).3. For \( x^{-5} \), \( n = -5 \), so the antiderivative is \( \frac{x^{-5+1}}{-5+1} = \frac{x^{-4}}{-4} = -\frac{x^{-4}}{4} \).

Step 4: Combine the Antiderivatives

Combine the antiderivatives of each term to form the complete antiderivative function:\[ F(x) = \frac{x^2}{2} + \frac{x^6}{6} - \frac{x^{-4}}{4} + C \]where \( C \) is the constant of integration.

Key Concepts

Indefinite Integral

The indefinite integral is a fundamental concept when finding antiderivatives. It represents the family of all antiderivatives of a function. Unlike definite integrals, which compute the area under a curve over a specific interval, an indefinite integral does not have limits of integration. Instead, it is expressed as \[ \int f(x) \, dx = F(x) + C \] where \( f(x) \) is the function being integrated, and \( F(x) \) is the antiderivative.

The result of an indefinite integral is a general function that includes an arbitrary constant, \( C \), due to the fact that differentiating a constant results in zero. This means any constant can be added to \( F(x) \) without affecting its derivative, providing an entire family of solutions.

When calculating an indefinite integral, always remember to include this constant of integration to ensure all possible solutions are considered.

Power Rule

The power rule is a critical technique when evaluating indefinite integrals. It is often used to find the antiderivatives of polynomial expressions because it simplifies complex algebraic functions effectively. The power rule states:

• For a term \( x^{n} \), the antiderivative is \( \frac{x^{n+1}}{n+1} + C \) where \( n eq -1 \).

This rule is particularly useful because it allows for straightforward integration of terms where the exponent is not equal to -1, breaking down the problem into manageable parts.

Let's look at an example: For the term \( x^{-5} \), applying the power rule, we find the antiderivative is:\[ \frac{x^{-5+1}}{-5+1} = -\frac{x^{-4}}{4} \]

By using the power rule, students can efficiently calculate antiderivatives and better understand how each term within a function contributes to the overall solution.

Constant of Integration

The constant of integration, denoted as \( C \), plays a crucial role in the solution of indefinite integrals. When we integrate a function, we lose specific information about constant values, as they do not affect the derivative. As a result, every antiderivative includes a constant term, ensuring all possible solutions are represented.

Why is this important?

• If we integrate the derivative of a function, we cannot determine the original function exactly without additional information. The constant \( C \) accounts for any vertical shifts in the function.
• The presence of \( C \) highlights that integration is an "undoing" process of differentiation that doesn't uniquely identify the original function.

For example, if the antiderivative is found to be \( \frac{x^2}{2} + \frac{x^6}{6} - \frac{x^{-4}}{4} \), then its indefinite integral should be expressed as\[ F(x) = \frac{x^2}{2} + \frac{x^6}{6} - \frac{x^{-4}}{4} + C. \] Including \( C \) ensures a complete family of all potential solutions, encapsulating infinite variations of the function that would yield the same derivative.