Problem 12 - 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: Identify the Function Components

The function provided is \( f(x) = x + x^5 + x^{-5} \). This includes three terms: a linear term \(x\), a polynomial term \(x^5\), and a negative power term \(x^{-5}\).

Step 2: Apply Power Rule for Antiderivatives

The power rule for finding antiderivatives states that to integrate \(x^n\), the antiderivative is \( \frac{x^{n+1}}{n+1} + C \), where \(neq -1\) and \(C\) is the constant of integration.

Step 3: Find the Antiderivative of Each Term

**For the first term \(x\):**\[ \int x \; dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2} \]**For the second term \(x^5\):**\[ \int x^5 \; dx = \frac{x^{5+1}}{5+1} = \frac{x^6}{6} \]**For the third term \(x^{-5}\):**\[ \int x^{-5} \; dx = \frac{x^{-5+1}}{-5+1} = \frac{x^{-4}}{-4} = -\frac{x^{-4}}{4} \]

Step 4: Combine the Antiderivatives

Now, combine each of the antiderivatives with the constant of integration, \(C\):\[ F(x) = \frac{x^2}{2} + \frac{x^6}{6} - \frac{x^{-4}}{4} + C \]

Key Concepts

Power Rule

The Power Rule is a fundamental concept in calculus used for finding antiderivatives. It's particularly simple yet immensely useful. This rule applies when you want to find the antiderivative of a term in the form of \(x^n\), where \(n\) is any real number except -1.

The essence of the Power Rule for antiderivatives is this: when you integrate \(x^n\), you increase the exponent by one and then divide by this new exponent. Mathematically, this is expressed as follows:

• For \(x^n\), the antiderivative is \( \frac{x^{n+1}}{n+1} + C \).
• The \(+C\) represents the constant of integration, which appears because when we differentiate a constant, its derivative is zero.

Breaking this down further, consider \(x\) as a term itself, which is \(x^1\). Applying the Power Rule here:

• First, increase the exponent 1 by 1, resulting in \(x^2\).
• Then divide by the new exponent: \( \frac{x^2}{2} \).

Remember, the Power Rule is inapplicable if \(n = -1\), as division by zero would occur. In this case, the antiderivative of \(x^{-1}\) is \(\ln |x|\).

Integration

Integration is the process of finding an antiderivative of a function. In simple terms, integrating a function gives you another function, which, when differentiated, takes you back to the original function.

The main purpose of integration in the context of antiderivatives is to solve equations involving derivatives, like finding areas under curves, or solving physical problems including displacement and accumulative change.

• Integration involves summing up infinitely many small pieces to determine a whole.
• The symbol \(\int\) is used to denote the process of integration.

When dealing with functions like polynomials, integration becomes straightforward as we apply the Power Rule. However, integration can become complex when we encounter functions that do not neatly fall into simple forms.

In practice, when integrating a sum of functions, we integrate each term separately. For instance, in this exercise, each term of the polynomial \(f(x) = x + x^5 + x^{-5}\) is integrated individually and only then are the results combined.

Polynomial Functions

Polynomial functions consist of variables raised to integer powers and coefficients. They can have terms like \(c_n x^n + c_{n-1} x^{n-1} + \, ... \, + c_1 x + c_0\), where \(c_n, c_{n-1}, ..., c_0\) are constants.

In the exercise, each term—\(x\), \(x^5\), and \(x^{-5}\)—is a part of the larger polynomial function. Such functions are particularly amenable to integration:

• The antiderivatives of polynomials are simply determined by applying the Power Rule individually to each term.
• This property makes polynomial functions one of the simplest types of functions to work with regarding integration.

Polynomial functions can also have negative exponents, as seen with \(x^{-5}\), although while polynomials typically feature only non-negative integer exponents, these terms can still be managed similarly with a keen understanding of the Power Rule.

These functions are everywhere in mathematics due to their straightforward nature and ability to model a wide range of real-world scenarios, highlighting the importance of mastering their integration.