Problem 23 - Thomas Calculus in SI Units

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. $$\int\left(\frac{1}{x^{2}}-x^{2}-\frac{1}{3}\right) d x$$

Step-by-Step Solution

Answer

The correct antiderivative is \( -\frac{1}{x} - \frac{x^{3}}{3} - \frac{1}{3}x + C \).

Step 1: Identify the Antiderivative Components

The integral \( \int \left( \frac{1}{x^{2}}-x^{2}-\frac{1}{3} \right) dx \) can be broken down into separate parts for which we can find antiderivatives individually: \( \frac{1}{x^{2}} \), \( -x^{2} \), and \( -\frac{1}{3} \).

Step 2: Find the Antiderivative of \( \frac{1}{x^{2}} \)

Recall that \( \frac{1}{x^{2}} = x^{-2} \). The antiderivative of \( x^{n} \) is \( \frac{x^{n+1}}{n+1} + C \), so \( \int x^{-2} \, dx = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C \).

Step 3: Find the Antiderivative of \( -x^{2} \)

For the term \( -x^{2} \), use the power rule: \( \int -x^{2} \, dx = -\frac{x^{3}}{3} + C \).

Step 4: Find the Antiderivative of \( -\frac{1}{3} \)

The integral of a constant \( c \) with respect to \( x \) is \( cx + C \). Thus, \( \int -\frac{1}{3} \, dx = -\frac{1}{3}x + C \).

Step 5: Combine the Antiderivatives

Combine the results from the previous steps to form the general antiderivative: \[ \int \left( \frac{1}{x^{2}} - x^{2} - \frac{1}{3} \right) dx = -\frac{1}{x} - \frac{x^{3}}{3} - \frac{1}{3}x + C \], where \( C \) is the constant of integration.

Step 6: Verify by Differentiation

Differentiate the result: if \( F(x) = -\frac{1}{x} - \frac{x^{3}}{3} - \frac{1}{3}x + C \), then \( F'(x) = \frac{d}{dx} \left( -\frac{1}{x} - \frac{x^{3}}{3} - \frac{1}{3}x \right) = \frac{1}{x^{2}} - x^{2} - \frac{1}{3} \), which matches the original function under the integral sign.

Key Concepts

Antiderivative

An antiderivative is a crucial concept in calculus, often referred to as the reverse of differentiation. When you differentiate a function, you find its derivative. When you find an antiderivative, you're essentially working backwards. The question asks us to find the antiderivative or indefinite integral of a specific expression. In simple terms, the goal is to determine a function whose derivative matches the expression given in the integral. To break it down:

• Think of differentiation as "finding the slope" of a function at any point.
• Antidifferentiation, or finding the antiderivative, is like looking for a function that produces a particular slope.

In the problem, the expression to integrate is \( \int \left( \frac{1}{x^{2}} - x^{2} - \frac{1}{3} \right) dx \). The given step-by-step solution wisely separates each term and finds the antiderivative individually. This approach makes it easier to handle the entire integral. Thus, understanding antiderivatives is essential for solving integrals and reconstructing potential original functions.

Power Rule

The power rule is a fundamental technique used to find integrals and derivatives. It states that if you have a power of x, the antiderivative can be found using a specific simple formula. For finding indefinite integrals, the power rule says: 1. Increase the exponent by 1.2. Divide by the new exponent.For example, if you have a term like \( x^n \), the antiderivative would be \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration, which we'll cover later. In our exercise:

• For \( \frac{1}{x^2} = x^{-2} \), the power rule changes it to \( -\frac{1}{x} \).
• For \( -x^2 \), applying the power rule results in \( -\frac{x^3}{3} \).
• For constants like \( -\frac{1}{3} \), integration involves multiplying by \( x \), resultingly \( -\frac{1}{3}x \).

The power rule simplifies integration patterns and makes solving integrals straightforward, primarily when dealing with polynomials.

Constant of Integration

The constant of integration, often symbolized as \( C \), plays a significant role in indefinite integrals. When solving an indefinite integral, you are essentially looking for all possible antiderivatives of a given function. Here's why it's important:

• When finding the derivative of any constant, the result is zero, which means when calculating antiderivatives, any constant could have been part of the original function.
• You add \( C \) to the antiderivative to account for this unknown constant factor.

In the problem \( \int \left( \frac{1}{x^{2}} - x^{2} - \frac{1}{3} \right) dx \), the "\( + C \)" at the end of each separately integrated part ensures that all potential vertical shifts of the antiderivative function are covered. This concept reflects the idea that there is a family of curves (functions) that can satisfy the original integral, and the constant \( C \) will adjust each member of this family vertically. Understanding the constant of integration helps grasp the full picture of indefinite integrals.