Problem 5 - Essential Calculus Early Transcendentals

Question

Find the most general antiderivative of the function (Check your answer by differentiation.) $$f(x)=3 \sqrt{x}-2 \sqrt[3]{x}$$

Step-by-Step Solution

Answer

The most general antiderivative is \( F(x) = 2x^{3/2} - \frac{3}{2}x^{4/3} + C \).

Step 1: Rewrite the function in simpler terms

The given function is \( f(x) = 3 \sqrt{x} - 2 \sqrt[3]{x} \). First, rewrite these radicals in terms of fractional exponents: \( f(x) = 3x^{1/2} - 2x^{1/3} \). This will make it easier to integrate.

Step 2: Apply the power rule for integration

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). Applying this rule, the antiderivative of \( 3x^{1/2} \) is \( 3 \cdot \frac{x^{3/2}}{3/2} = 2x^{3/2} \), and the antiderivative of \( -2x^{1/3} \) is \( -2 \cdot \frac{x^{4/3}}{4/3} = -\frac{6}{4}x^{4/3} = -\frac{3}{2}x^{4/3} \).

Step 3: Combine the results

Combine the individual antiderivatives to form the most general antiderivative of the original function: \( F(x) = 2x^{3/2} - \frac{3}{2}x^{4/3} + C \). Add the constant \( C \) to signify the general form of the antiderivative.

Step 4: Verify by differentiation

Differentiate \( F(x) = 2x^{3/2} - \frac{3}{2}x^{4/3} + C \) to check if we obtain \( f(x) \) again. Using the power rule for differentiation, \( \frac{d}{dx} [2x^{3/2}] = 3x^{1/2} \) and \( \frac{d}{dx} [-\frac{3}{2}x^{4/3}] = -2x^{1/3} \). The derivative of the constant \( C \) is zero. Thus, \( F'(x) = 3x^{1/2} - 2x^{1/3} = f(x) \).

Key Concepts

Power Rule for Integration

When we integrate functions, one handy tool is the power rule for integration. This rule is quite simple yet powerful. It helps us find antiderivatives of power functions. Here's what it says: if you have a function of the form \(x^n\), then its integral is \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \( n eq -1 \).
This formula means you increase the exponent by 1, and then divide by this new exponent. The \( C \) is a constant of integration, representing all possible vertical shifts of the antiderivative.
For example, integrating \( 3x^{1/2} \) using the power rule, you find \( 3 \cdot \frac{x^{3/2}}{3/2} = 2x^{3/2} \). When you integrate \( -2x^{1/3} \), use \( -2 \cdot \frac{x^{4/3}}{4/3} = -\frac{3}{2}x^{4/3} \). Integrating step by step with this rule simplifies calculations and ensures accurate results each time.

Fractional Exponents

Fractional exponents are quite useful in calculus, especially when dealing with roots of numbers. They transform root-related expressions into exponent forms, making them easier to handle with differentiation and integration.

• \( \sqrt{x} \) is expressed as \( x^{1/2} \)
• \( \sqrt[3]{x} \) is written as \( x^{1/3} \)

By rewriting expressions with fractional exponents, you can apply rules that deal with powers directly.
For instance, converting \( 3\sqrt{x} - 2\sqrt[3]{x} \) into \( 3x^{1/2} - 2x^{1/3} \) allows us to use the power rule for integration efficiently. These reforms help streamline solving processes, providing a consistent approach to dealing with roots and powers.

Differentiation Verification

After finding an antiderivative, it's crucial to verify its correctness using differentiation. This involves checking that when you take the derivative of your result, it matches the original function you started with.
To verify, differentiate the antiderivative step-by-step:

• Apply the power rule for differentiation, which states that \( \frac{d}{dx} [x^n] = nx^{n-1} \).
• For our function, differentiating \( 2x^{3/2} \) gives \( 3x^{1/2} \).
• Similarly, \( \frac{d}{dx} [-\frac{3}{2}x^{4/3}] = -2x^{1/3} \).

The constant \( C \) disappears because the derivative of a constant is zero.
After differentiation, if the result matches the original function \( f(x) \), then the antiderivative is confirmed correct. This step is essential because it validates our integration process and ensures that our antiderivative is indeed accurate.