Problem 11 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=1+\frac{1}{x^{2}} $$

Step-by-Step Solution

Answer

The general antiderivative of \( f(x) = 1 + \frac{1}{x^2} \) is \( F(x) = x - \frac{1}{x} + C \).

Step 1: Recognize Function Components

The function given is \( f(x) = 1 + \frac{1}{x^2} \). This can be broken down into two parts: a constant function, \( 1 \), and a power function, \( \frac{1}{x^2} = x^{-2} \). We need to find the antiderivative of each component separately.

Step 2: Antiderivative of Constant

The antiderivative of the constant \( 1 \) is \( x \). This is because the derivative of \( x \) with respect to \( x \) is \( 1 \).

Step 3: Antiderivative of Power Term

To find the antiderivative of \( x^{-2} \), use the power rule for antiderivatives: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \). Here, \( n = -2 \), so the antiderivative is \( \frac{x^{-1}}{-1} = -x^{-1} = -\frac{1}{x} \).

Step 4: Combine the Results

Now, combine the antiderivatives of the two components to find the overall antiderivative: \( F(x) = x - \frac{1}{x} + C \), where \( C \) is the constant of integration.

Key Concepts

Power Rule

Understanding the power rule is crucial when dealing with antiderivatives, especially when it comes to polynomial and power functions. The power rule for integration is a straightforward method for finding antiderivatives of functions of the form \( x^n \). The rule states:

• For a function \( x^n \), the antiderivative is given by \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \) where \( n eq -1 \).

This formula is derived from the derivative rules, essentially reversing the process.
For example, if \( f(x) = x^{-2} \), apply the power rule:
1. Increase the exponent by 1, resulting in \( n+1 = -2 + 1 = -1 \).2. Divide by the new exponent to get \(-\frac{x^{-1}}{1} = -x^{-1} \).
In cases where \( n = -1 \), the antiderivative is known to be the natural logarithm, \( \ln|x| \), but this doesn’t apply to our current function \( f(x) = 1 + \frac{1}{x^2} \).

Integration Constant

When integrating functions to find antiderivatives, you must remember to include the constant of integration, often represented by \( C \). This constant is essential because integration is the reverse of differentiation, and when differentiating a constant, it vanishes (since the derivative of a constant is zero).
So when finding the antiderivative, to account for any constant that might have been part of the original function before it was differentiated, we add \( C \) to the result. This ensures we represent the entire family of functions that could have the same derivative.

• For instance, the antiderivative of \( 1 \) is \( x + C \).
• Similarly, for \( \frac{1}{x^2} \) which becomes \( -\frac{1}{x} + C \).

Adding these together, the general antiderivative of our function \( f(x) = 1 + \frac{1}{x^2} \) is \( F(x) = x - \frac{1}{x} + C \).

Calculus Problem Solving

Solving calculus problems often involves systematically applying various rules and techniques. Antiderivatives require a reverse approach compared to derivatives. Instead of finding the rate of change, you're assessing the accumulated area under the curve or a previous point in the function's evolution.
Effective problem-solving involves:

• Carefully breaking down the given function into components.
• Identifying applicable rules, such as the power rule for individual parts.
• Remembering to include the integration constant \( C \).

In our problem with \( f(x) = 1 + \frac{1}{x^2} \), we broke it down into two easier parts: \( 1 \) and \( x^{-2} \). Each part was dealt with using its antiderivative rule, demonstrating a strategic approach to problem-solving in calculus.

Function Decomposition

Decomposing a function is a powerful technique in calculus that simplifies complex problems into more manageable parts. This method involves breaking down a composite function into its individual components, which can be solved separately.
For the given function \( f(x) = 1 + \frac{1}{x^2} \), we identified it as a sum of two simpler functions: a constant \( 1 \), and a power term \( \frac{1}{x^2} \) or equivalently \( x^{-2} \). Each piece was then integrated individually.

• The constant \( 1 \) has a known antiderivative, \( x \).
• The power term \( \frac{1}{x^2} \) can be rewritten as \( x^{-2} \), where the power rule applies.

Function decomposition not only simplifies the antiderivative process but also makes it easier to identify errors and verify results. This step-by-step reduction is crucial for tackling more complex integration problems in calculus.