Problem 8 - Thomas Calculus

Question

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }\frac{4}{3} \sqrt[3]{x} \quad \text { b. } \frac{1}{3 \sqrt[3]{x}} \quad \text { c. } \sqrt[3]{x}+\frac{1}{\sqrt[3]{x}}$$

Step-by-Step Solution

Answer

a. \( x^{4/3} + C \), b. \( \frac{1}{2} x^{2/3} + C \), c. \( \frac{3}{4} x^{4/3} + \frac{3}{2} x^{2/3} + C \)

Step 1: Understanding Antiderivative

The antiderivative of a function is the inverse process of differentiation. For each part of the problem, the goal is to find a function whose derivative is the given function.

Step 2: Solve Part a

Given the function \( f(x) = \frac{4}{3} \sqrt[3]{x} = \frac{4}{3} x^{1/3} \), we need to find its antiderivative. Use the power rule for integration, which states that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \). Applying this, the antiderivative will be: \[ \int \frac{4}{3} x^{1/3} \, dx = \frac{4}{3} \cdot \frac{x^{4/3}}{4/3} + C = x^{4/3} + C \] where \( C \) is the constant of integration.

Step 3: Solve Part b

Given the function \( f(x) = \frac{1}{3 \sqrt[3]{x}} = \frac{1}{3} x^{-1/3} \), use the integration power rule. The antiderivative is: \[ \int \frac{1}{3} x^{-1/3} \, dx = \frac{1}{3} \cdot \frac{x^{2/3}}{2/3} + C = \frac{1}{2} x^{2/3} + C \] where \( C \) is the constant of integration.

Step 4: Solve Part c

Given the function \( f(x) = \sqrt[3]{x} + \frac{1}{\sqrt[3]{x}} = x^{1/3} + x^{-1/3} \), split the integration and apply the power rule to each term: \[ \int (x^{1/3} + x^{-1/3}) \, dx \] The antiderivative of each part is: \[ \frac{x^{4/3}}{4/3} + \frac{x^{2/3}}{2/3} + C = \frac{3}{4} x^{4/3} + \frac{3}{2} x^{2/3} + C \] where \( C \) is the constant of integration.

Step 5: Verify by Differentiation

To ensure correctness, differentiate each antiderivative to check if it corresponds to the original function. For example, differentiating \( x^{4/3} \) gives \( \frac{4}{3} x^{1/3} \), confirming it's correct for part a. Repeat similar checks for parts b and c.

Key Concepts

Integration Power Rule

The integration power rule is a key tool in the process of finding antiderivatives. It is particularly useful when dealing with functions that are expressed as powers of the variable, such as those in polynomial form. The rule states that the antiderivative of a function of the form \( x^n \) is given by:

\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

where \( n eq -1 \) and \( C \) is the constant of integration, which we'll discuss later. This formula is the reverse of the differentiation power rule.

• Apply this rule separately to each term in a polynomial.
• Remember, each term integrates independently.
• Always add the constant of integration \( C \) to account for any possibilities in the family of antiderivatives.

For example, if you have \( \frac{4}{3} x^{1/3} \), you increase the exponent by one to get \( x^{4/3} \) and divide by the new exponent, then multiply by any constants present. This process is applied to each term in a polynomial for complete integration.

Differentiation Verification

Once you have found an antiderivative of a function using integration, it is crucial to check your work. Differentiation verification allows you to confirm that the antiderivative you found is correct. Essentially, you do this by differentiating the antiderivative:

If you differentiate \( F(x) \), which is the antiderivative of \( f(x) \), and retrieve \( f(x) \), then \( F(x) \) is the correct antiderivative.

Let's take a look at how this works with an example:
For the antiderivative \( \int \frac{4}{3} x^{1/3} \, dx = x^{4/3} + C \), differentiate \( x^{4/3} + C \):

• Find the derivative of \( x^{4/3} \). This utilizes the power rule of differentiation, which is \( \frac{d}{dx} x^n = n x^{n-1} \).
• This gives us \( \frac{4}{3} x^{1/3} \), which is exactly the function we began with.

This confirms that our integration was done correctly. Each part of a problem can be verified similarly, ensuring each antiderivative aligns back to its original function.

Constant of Integration

The constant of integration, often denoted as \( C \), represents an important aspect of indefinite integrals. Unlike definite integrals that calculate a specific number, indefinite integrals signify a family of functions. This is because when you differentiate a constant, it vanishes, meaning when finding the antiderivative, any constant value could have been there initially and would have differentiated to zero.
This constant is what differentiates indefinite integration from other forms of calculus, and it plays a major role in solutions to differential equations.

• Whenever you integrate, always include \( C \) to represent all potential vertical shifts of the function.
• Without this constant, the solution would lack generality, possibly leading to incorrect assumptions when solving physics or engineering problems.
• \( C \) is critical in adjusting the antiderivative to fit specific initial conditions or constraints.

Thus, the constant of integration preserves the essence of what makes integration so powerful—its ability to express multiple possibilities within one streamlined solution.