Problem 16 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=x^{7}+\frac{1}{x^{7}} $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = \frac{x^8}{8} - \frac{1}{6x^6} + C \).

Step 1: Identify Each Term

The function given is composed of two separate terms: \( x^7 \) and \( \frac{1}{x^7} \). Recognize that the general antiderivative involves finding the antiderivative for each term independently.

Step 2: Apply Power Rule to First Term

For the first term \( x^7 \), use the power rule for integration which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) where \( n eq -1 \). Therefore, \( \int x^7 \, dx = \frac{x^{8}}{8} + C_1 \).

Step 3: Rewrite and Apply Power Rule to Second Term

Rewrite the second term \( \frac{1}{x^7} \) as \( x^{-7} \). Again, apply the power rule: \( \int x^{-7} \, dx = \frac{x^{-6}}{-6} + C_2 \).

Step 4: Combine Antiderivatives

Combine the two antiderivatives. The general antiderivative of \( f(x) = x^7 + \frac{1}{x^7} \) is: \( F(x) = \frac{x^8}{8} - \frac{x^{-6}}{6} + C \), where \( C = C_1 + C_2 \) is the constant of integration.

Key Concepts

Power Rule in Integration

The Power Rule is a fundamental principle in calculus, especially useful for finding antiderivatives, which are essentially the reverse of derivatives. It helps us simplify the integration of polynomial functions. To use the Power Rule when finding an antiderivative:

• Identify the exponent of the variable in each term.
• Apply the rule: if you have a term in the form of \( x^n \), the antiderivative is \( \frac{x^{n+1}}{n+1} \) plus a constant \( C \).
• Keep in mind, the rule is only valid when \( n eq -1 \) because division by zero is undefined.

For example, to find the antiderivative of \( x^7 \):

• Increase the exponent by one to get 8,
• Divide by the new exponent: \( \frac{x^8}{8} \).

The Power Rule simplifies the problem, breaking it down into a straightforward series of steps.

Understanding Integration

Integration is the process of finding the antiderivative, often referred to as the area under the curve of a function. It reverses differentiation and gives us the original function given its rate of change. In this context, integration is not just a mechanical process but a way of identifying the accumulation of quantities:

• Definite Integration: Involves calculating the exact area under a curve between two bounds.
• Indefinite Integration: As seen here, seeks the general antiderivative without specific limits, hence the solution is expressed with a constant \( C \).

When integrating a function like \( f(x) = x^7 + \frac{1}{x^7} \):

• Separate the function into individual terms.
• Apply appropriate integration rules, such as the Power Rule, to each term independently.

By integrating, we uncover the primitive function whose derivative corresponds to the original function, providing deeper insights into its behavior.

Role of the Constant of Integration

When finding an antiderivative using indefinite integration, you always include a constant of integration, denoted as \( C \). This constant is crucial because:

• Differentiation removes constants, meaning the original function could have had any constant added.
• Constant \( C \) represents an infinite number of possible vertical shifts of the antiderivative graph.
• It ensures that the set of all possible antiderivatives are accurately represented.

For example, if you have an antiderivative \( F(x) = \frac{x^8}{8} - \frac{x^{-6}}{6} \) and no other context, \( C \) accounts for any potential shifts in \( F(x) \). This inclusion respects the mathematical principle of integration as it ensures completeness by covering all possible original functions.