Problem 33 - University Calculus: Early Transcendentals

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int x^{-1 / 3} d x$$

Step-by-Step Solution

Answer

\( \int x^{-1/3} \, dx = \frac{3}{2} x^{2/3} + C \).

Step 1: Identify the Power Rule for Integration

The integral we need to solve is \( \int x^{-1/3} \, dx \). To integrate a function of the form \( x^n \), we use the power rule for integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \).

Step 2: Apply the Power Rule

For our integral \( \int x^{-1/3} \, dx \), the exponent \( n = -1/3 \). According to the power rule, the antiderivative is \( \frac{x^{(-1/3) + 1}}{-1/3 + 1} + C \).

Step 3: Simplify the Expression

Calculate \( (-1/3) + 1 = 2/3 \). So, our expression becomes \( \frac{x^{2/3}}{2/3} + C \). Simplify by multiplying by the reciprocal of \( 2/3 \): \( \frac{1}{2/3} = \frac{3}{2} \). Therefore, the expression simplifies to \( \frac{3}{2} x^{2/3} + C \).

Step 4: Verify by Differentiation

Differentiate \( \frac{3}{2} x^{2/3} + C \) to verify. The derivative of \( \frac{3}{2} x^{2/3} \) is \( \frac{3}{2} \times \frac{2}{3} x^{(2/3) - 1} = x^{-1/3} \). The constant \( C \) differentiates to 0. The derivative matches the original integrand, confirming the solution.

Key Concepts

Integration

In calculus, integration is a core concept that deals with finding the integral or anti-derivative of a function. Think of integration as the reverse process of differentiation. If differentiation tells us the rate at which something changes, integration helps us find the original function before any changes happened.

The symbol for integration is \int, and it represents the process of accumulating values, such as area under a curve. The result of an integration process is called the integral. There are two types of integrals: definite and indefinite integrals.

• **Definite Integral**: Computes the exact area under a curve between two bounds, producing a set numerical value.
• **Indefinite Integral**: Finds the most general antiderivative of a function, including a constant of integration, often represented by \(C\). It describes a family of functions.

For example, when solving \int x^{-1/3} \, dx, we are looking for the antiderivative of \ x^{-1/3}. Integration steps through various methods depending on functions involved. For many, the power rule is a starting point.

Antiderivatives

An antiderivative is a function whose derivative is the original function. In simpler terms, it’s the reverse of finding a derivative. If we differentiate an antiderivative, we get back to our original function. This is often explored through integration.

Each function can have many antiderivatives. That’s because integration introduces a constant term, \(C\), which can take any value. This infinite nature is seen in the family of curves generated when you find an antiderivative.

• **Example:** The antiderivative of \ 2x \ is \ x^2 + C\. Differentiating \ x^2 + C\ returns \ 2x \, illustrating that \ x^2 + C\ forms a family of solutions.

Equipping ourselves with the ability to reverse engineer a given function to find its original form helps in numerous applications, from physics to engineering, where understanding the base phenomenon can reveal much about the data being analyzed.

Power Rule for Integration

The Power Rule for Integration is a basic yet powerful tool used to find the antiderivatives of polynomial functions. This rule is straightforward: if you can express the function in the form \( x^n \), the integral becomes \int x^n \, dx = \frac{x^{n+1}}{n+1} + C\, given that \ n \ is not -1.

When using the Power Rule to integrate \ x^{-1/3} \, as in our example problem, we follow these steps:

• **Adjust the Exponent:** Identify that \ n \ is \ -1/3\. Using the power rule, adjust the exponent by adding 1: \ (-1/3) + 1 = 2/3\.
• **Calculate the Coefficient:** Divide by the new exponent, give: \ \frac{1}{2/3} = \frac{3}{2} \ for the coefficient.
• **Combine and Simplify:** The result is \ \frac{3}{2} x^{2/3} + C \.

The rule simplifies the process of finding antiderivatives, making it a go-to method in many integration problems. Most importantly, verifying the derived function through differentiation gives reassurance that the integration was carried out correctly.