Problem 6 - Calculus

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

Step 1: Identify the function

The function given is \( f(x) = 3x^{2/3} \). We need to find its antiderivative, which involves integrating the function.

Step 2: Recall the Power Rule for Integration

The power rule for integration tells us that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} \) if \( n eq -1 \). We can apply this rule to integrate \( 3x^{2/3} \).

Step 3: Apply the Power Rule

Apply the power rule to the function: \[ \int 3x^{2/3} \, dx = 3 \cdot \frac{x^{2/3 + 1}}{2/3 + 1} \] Calculate \( 2/3 + 1 = 5/3 \).

Step 4: Simplify the Expression

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

Step 5: Add the Constant of Integration

Since we are finding the general antiderivative, we need to add the constant of integration \( C \). Thus, the antiderivative of \( f(x) = 3x^{2/3} \) is \[ F(x) + C = \frac{9}{5}x^{5/3} + C \].

Key Concepts

Power Rule for Integration

The Power Rule for Integration is a fundamental technique used for finding antiderivatives of polynomial expressions. This rule is particularly useful because it simplifies the integration process by directly applying a formula. The basic idea is straightforward: if you have a function in the form of \( x^n \), where \( n \) is not equal to \(-1\), the antiderivative is given by:

• \( \frac{x^{n+1}}{n+1} \)

This formula helps us "undo" the differentiation of power functions.

In the given exercise, the function is \( f(x) = 3x^{2/3} \). By applying the power rule, we calculate \( n + 1 \) where \( n = \frac{2}{3} \), yielding \( \frac{5}{3} \). The antiderivative becomes \( \frac{x^{5/3}}{5/3} \). This step transforms the complex-looking power into a manageable expression, allowing further simplification.

Constant of Integration

When finding antiderivatives, always remember to add the Constant of Integration, denoted as \( C \). This constant represents an infinite number of possible vertical shifts of the antiderivative.

The reason behind this is simple: differentiation eliminates constants. So when you differentiate any constant, its derivative is zero. When we integrate and recover the original function, we can miss any constant that was there initially. Therefore, the general solution to an indefinite integral should always include \( C \).

• Example: \( \,F(x) + C = \frac{9}{5}x^{5/3} + C\)

Incorporating the constant of integration is crucial in expressing the full range of solutions an antiderivative can provide.

Integration Technique

Solving the given function \( f(x) = 3x^{2/3} \) involves selecting the correct integration technique. For polynomial-like expressions or functions that can be decomposed into powers of x, applying the Power Rule is efficient.

Here's a detailed step-by-step approach:

• Identify if the function is suitable for the Power Rule by recognizing its polynomial features.
• Adjust the power by incrementing it by 1, which is necessary for integration.
• Divide by this new power to complete the antiderivative formula \( \frac{x^{n+1}}{n+1} \).
• Simplify the expression as needed for easier understanding and applications, transforming fractions and constants if required.

In this exercise, we took \( 3x^{2/3} \) and used the Power Rule to find the antiderivative. Practice with various functions will show how powerful and versatile this integration technique really is.