Problem 13 - Essential Calculus Early Transcendentals

Question

Find the most general antiderivative of the function (Check your answer by differentiation $$f(x)=\frac{x^{5}-x^{3}+2 x}{x^{4}}$$

Step-by-Step Solution

Answer

The most general antiderivative is \( F(x) = \frac{x^2}{2} - \ln|x| - x^{-2} + C \).

Step 1: Simplify the Function

First, simplify the function given: \( f(x) = \frac{x^{5} - x^{3} + 2x}{x^{4}} \). We can simplify it by dividing each term in the numerator by the denominator. This gives: \[ f(x) = x + \frac{-x^{3}}{x^{4}} + \frac{2x}{x^{4}} = x - x^{-1} + 2x^{-3}. \]

Step 2: Find the Antiderivative of Each Term

The antiderivative of a sum of functions is the sum of their antiderivatives. Therefore, we find the antiderivative of each term individually. For \( x \), the antiderivative is \( \frac{x^2}{2} + C_1 \). For \( -x^{-1} \), the antiderivative is \( -\ln|x| + C_2 \). For \( 2x^{-3} \), the antiderivative is \( -x^{-2} + C_3 \). There will be one arbitrary constant for the whole expression, so we can denote it as just \( C \).

Step 3: Combine the Antiderivatives

Combine the antiderivatives you calculated: \[ F(x) = \frac{x^2}{2} - \ln|x| - x^{-2} + C. \] This expression represents the most general antiderivative of the given function \( f(x) \).

Step 4: Verify the Antiderivative by Differentiation

Differentiate \( F(x) = \frac{x^2}{2} - \ln|x| - x^{-2} + C \) to verify. The derivative of \( \frac{x^2}{2} \) is \( x \), the derivative of \( -\ln|x| \) is \( -x^{-1} \), and the derivative of \( -x^{-2} \) is \( 2x^{-3} \). Constants differentiate to zero. Therefore: \[ \frac{d}{dx}F(x) = x - x^{-1} + 2x^{-3} = f(x), \] which confirms our antiderivative is correct.

Key Concepts

Understanding Calculus

Calculus is a branch of mathematics that helps us understand how things change. It revolves around two main concepts: differentiation, which deals with rates of change, and integration, which deals with the accumulation of quantities.
This subject is used to solve problems related to motion, space, time, and much more. It forms the basis for advanced topics in mathematics, science, and engineering.
In most practical applications, you use calculus to understand and model real-world situations, such as predicting the spread of diseases, optimizing operations, and even finding the shortest path between two points. Understanding the foundational principles of calculus is essential for students aiming to succeed in these fields.

Exploring Integration

Integration is one of the core operations in calculus, working as the opposite of differentiation. While differentiation tells you at what rate something is changing, integration gives you the accumulated quantity over time or space.
The process of integration can be understood as finding the "antiderivative" of a function. For example, in the exercise, you find the antiderivative of a simplified function.

• The purpose of integration is to calculate areas under curves, find volumes, and determine related rates in various contexts.
• There are different techniques for integration, such as substitution, integration by parts, and partial fraction decomposition.
• Each method simplifies the integration process, making it easier to solve more complex problems.

By practicing integration, you develop problem-solving skills essential for tackling intricate mathematical and real-world challenges.

Clarifying Differentiation

Differentiation is a fundamental concept in calculus that focuses on finding the rate of change of a quantity. It helps in understanding how a function's output changes as its input changes. In simple terms, differentiation gives you the function's derivative.
To differentiate, you use rules such as the power rule, product rule, and chain rule. In the step-by-step solution, differentiation is employed to verify the antiderivative function.

• The power rule, for example, states that the derivative of \( x^n \) is \( nx^{n-1} \).
• It's crucial for confirming that you correctly integrated the function in the prior steps.
• Additionally, differentiation is widely used in optimization problems where you need to find minimum or maximum values.

By mastering differentiation, you lay a strong foundation for solving dynamic problems, whether in mathematics or its countless applications in different fields.