Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = \frac{9}{5}x^{5/3} + C \).

Step 1: Understanding the Problem

We need to find the general antiderivative of the function \( f(x) = 3x^{2/3} \). This means we need a function \( F(x) \) such that \( F'(x) = f(x) \).

Step 2: Applying the Power Rule for Integration

The antiderivative of \( x^n \) is given by \( \frac{x^{n+1}}{n+1} \), provided \( n eq -1 \). For \( f(x) = 3x^{2/3} \), set \( n = \frac{2}{3} \).

Step 3: Calculating the Antiderivative

Using the rule from the previous step, integrate \( 3x^{2/3} \):\[ F(x) = 3 \cdot \frac{x^{2/3 + 1}}{2/3 + 1} = 3 \cdot \frac{x^{5/3}}{5/3} \]

Step 4: Simplifying the Expression

Simplify the expression:\[ F(x) = 3 \cdot \frac{3}{5}x^{5/3} = \frac{9}{5}x^{5/3} \]

Step 5: Adding the Constant of Integration

The general antiderivative includes a constant of integration, \( C \), so:\[ F(x) = \frac{9}{5}x^{5/3} + C \] where \( C \) is an arbitrary constant.

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental theorem in calculus that helps in finding the antiderivative or the general integral of power functions. This technique is related to the reverse process of differentiation, which is also known as integration. The rule states that to find the antiderivative of a function of the form \(x^n\), you can use the formula:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

This formula applies as long as \(n eq -1\). It's important to note that while the power rule for differentiation is a method for reducing exponents, integration effectively increases the exponent. Additionally, don't forget to simplify the expression after applying this rule, as shown in our example with \( f(x) = 3x^{2/3} \). By applying the power rule correctly, you obtain the antiderivative: \( F(x) = \frac{9}{5}x^{5/3} \).
Always remember that integration reverses differentiation, so reviewing both processes will help solidify the understanding of how y-values change with respect to x.

General Antiderivative

The general antiderivative of a function is, essentially, the reverse operation of differentiation. When you find the antiderivative, you are looking for a function whose derivative is the original function given.In the exercise, we started with \( f(x) = 3x^{2/3} \). To find its antiderivative, we used the power rule for integration, which led us to indentify the function \( F(x) = \frac{9}{5}x^{5/3} + C \). Here, \( F(x) \) is a family of functions and represents the general solution to the antiderivative problem. Each function in this family is identical except for a constant, which differentiates them.
By finding the general antiderivative, a wide variety of functions with different constants can represent the same rate of change found in the original \(f(x)\) function. This concept is significant because it's not only about a single solution but an entire set of potential solutions differing by the constant \( C \).
Understanding general antiderivatives enables you to solve many real-world problems that deal with accumulated quantities and can greatly enhance your ability to handle complex mathematical functions.

Constant of Integration

The constant of integration, denoted as \( C \), plays an important role when finding antiderivatives. Whenever you perform indefinite integration, you are actually looking for all possible functions whose derivative is the original function.After integrating a mathematical expression, it is crucial to add \( C \) to represent this infinite number of possible antiderivatives. For example, in the solution \( F(x) = \frac{9}{5}x^{5/3} + C \), this \( C \) accounts for variations in functions that could still result in the same derivative \( f(x) = 3x^{2/3} \). Without \( C \), you would miss these potential functions, and the solution would be incomplete.
The constant can represent various physical or geometrical properties depending on the context, such as initial conditions or boundary values in physical sciences. It reminds us that integration is about accumulation of values rather than a static process, and helps ensure that all aspects of change are considered.
Being comfortable with this concept is key in many areas of mathematics, from pure theory to practical applications in engineering and the sciences.