Problem 12 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=x^{3}-\frac{1}{x^{3}} $$

Step-by-Step Solution

Answer

The general antiderivative is \( \frac{x^4}{4} + \frac{x^{-2}}{2} + C \).

Step 1: Identify the Function Components

The function given is \( f(x) = x^3 - \frac{1}{x^3} \), which consists of two terms: \( x^3 \) and \( -\frac{1}{x^3} \). The task is to find the antiderivative of each term separately.

Step 2: Antiderivative of the First Term

The first term is \( x^3 \). The antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \) where \( n = 3 \). Thus, the antiderivative of \( x^3 \) is \( \frac{x^{4}}{4} \).

Step 3: Transform the Second Term

The second term is \( -\frac{1}{x^3} \), which can be rewritten as \( -x^{-3} \). This forms an exponent that allows us to apply the power rule for integration.

Step 4: Antiderivative of the Second Term

Using the power rule, the antiderivative of \( -x^{-3} \) is \( -\frac{x^{-2}}{-2} \). Simplifying this expression gives \( \frac{x^{-2}}{2} \).

Step 5: Combine Antiderivatives and Include Constant

The general antiderivative of the given function \( f(x) = x^3 - \frac{1}{x^3} \) is the sum of the two antiderivatives found: \( \frac{x^4}{4} + \frac{x^{-2}}{2} + C \), where \( C \) is the constant of integration.

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental concept in calculus that helps us find the antiderivative of functions in the form of power terms. It states that the antiderivative of any function in the form of \( x^n \) is \( \frac{x^{n+1}}{n+1} \), provided \( n eq -1 \). This rule simplifies the process of finding antiderivatives by adding 1 to the exponent and dividing by the new exponent.

For example, in the exercise, we have two terms: \( x^3 \) and \( -x^{-3} \). Applying the power rule, the antiderivative of \( x^3 \) becomes \( \frac{x^4}{4} \) because we add 1 to the exponent (3) making it 4, and divide by the same value (4).

For the term \( -x^{-3} \), it follows the same principle. The exponent increase gives us \( -x^{-2} \), and dividing by \(-2\) results in \( \frac{x^{-2}}{2} \). Both terms demonstrate how the power rule effectively simplifies integration when dealing with powers of \( x \).

Integration Constant

When finding the antiderivative, or the indefinite integral, of a function, an important inclusion is the integration constant, typically denoted as \( C \). The integration constant arises because taking the derivative of a constant is zero, meaning that multiple functions can have the same derivative.

In the context of our exercise, after we find the antiderivatives of the respective terms, we combine them to form the general antiderivative of the original function. This results in \( \frac{x^4}{4} + \frac{x^{-2}}{2} + C \). Here, \( C \) represents any constant value which could have been part of the original function but was eliminated in the differentiation process.

This inclusion ensures that our antiderivative solution accounts for all possible original functions whose derivative is \( f(x) = x^3 - \frac{1}{x^3} \). It's a crucial step in completing the integration process for indefinite integrals.

Polynomial Integration

Polynomial integration refers to the process of finding the antiderivative of polynomial expressions, which are sums of power terms like \( x^n \). Polynomials often present a straightforward application of the power rule, making them easier to integrate.

In the original exercise, the function \( f(x) = x^3 - \frac{1}{x^3} \) can be viewed through the lens of polynomial integration. Despite one term having a negative exponent, each part can still be handled using the same principle as polynomial integration: apply the power rule to each term individually and integrate.

Once every term is integrated, the resulting parts are summed. The antiderivative of our function thus becomes \( \frac{x^4}{4} + \frac{x^{-2}}{2} + C \). This highlights how polynomial integration systematically breaks down complex expressions into manageable parts, facilitating easier manipulation and understanding of the calculus involved.