Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=1 / \sqrt[3]{x^{2}} $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = 3x^{1/3} + C \).

Step 1: Understand the Function

We need to find the general antiderivative (also known as the indefinite integral) of the function given as \( f(x) = \frac{1}{\sqrt[3]{x^{2}}} \). This can be rewritten using exponent rules: \( f(x) = x^{-2/3} \).

Step 2: Apply the Power Rule for Antiderivatives

The power rule for antiderivatives states that for any function of the form \( x^n \), its antiderivative is \( \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). In this case, \( n = -\frac{2}{3} \). We apply this rule to find the antiderivative of \( x^{-\frac{2}{3}} \).

Step 3: Calculate the Antiderivative

Using the power rule, the antiderivative of \( x^{-\frac{2}{3}} \) becomes:\[F(x) = \frac{x^{-\frac{2}{3} + 1}}{-\frac{2}{3} + 1} + C = \frac{x^{\frac{1}{3}}}{\frac{1}{3}} + C = 3x^{\frac{1}{3}} + C\]

Step 4: Verify the Solution

Differentiate the antiderivative \( F(x) = 3x^{\frac{1}{3}} + C \) to ensure it results in the original function. Differentiating gives:\( F'(x) = 3 \cdot \frac{1}{3} x^{\frac{1}{3} - 1} = x^{-rac{2}{3}} \)This matches the original function, confirming our solution is correct.

Key Concepts

Indefinite Integrals

The concept of an indefinite integral is fundamental in calculus. It represents a family of functions whose derivative is the given function. When we solve an indefinite integral, we are essentially finding what original function, when differentiated, returns the function we started with. This means finding a formula that encompasses more solutions than just a single one because it includes an arbitrary constant, denoted as \(C\).

• The indefinite integral is often referred to as the antiderivative of a function.
• This family of functions is represented as \(F(x) + C\), where \(C\) can be any constant value.
• For example, if you find \(\int x^2 \, dx = \frac{x^3}{3} + C\), any value of \(C\) makes this answer correct.

In practice, indefinite integrals are used to solve problems in physics, engineering, and many other fields where you need to reconstruct a function from its rate of change or derivative.

Power Rule for Integration

The power rule for integration is a straightforward and essential rule for finding the antiderivative of polynomials and power functions. It simplifies the process of integration by providing a specific formula to use whenever you encounter powers of \(x\). The power rule states that for any power function \( x^n \), its antiderivative is given:\[\int x^n \,dx = \frac{x^{n+1}}{n+1} + C, \quad \text{where } n eq -1\]

• To apply the rule, add 1 to the exponent, then divide by the new exponent.
• Remember, this rule does not work when \(n = -1\), as it would make the denominator zero.
• An example using the power rule: \(\int x^4 \, dx\) becomes \(\frac{x^{5}}{5} + C\).

This rule is incredibly useful for integrating terms in equations and quickly finding solutions to many calculus problems.

Antiderivatives

Antiderivatives, often synonymous with indefinite integrals, are a core element in calculus with the primary focus of reversing differentiation. When we find the antiderivative of a function, we are trying to identify the original function that, when it is differentiated, results in the given function.

• Antiderivatives include a constant term \(C\), representing the infinite number of possible solutions.
• If \(F(x)\) is an antiderivative of \(f(x)\), then \(F'(x) = f(x)\).
• For example, the antiderivative of \(x^2\) is \(\frac{x^3}{3} + C\), meaning \(\left(\frac{x^3}{3}\right)' = x^2\).

Understanding antiderivatives is crucial for solving differential equations and analyzing the accumulation of quantities, which is a common task in both mathematics and applied sciences.