Problem 31 - 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. $$\int\left(\frac{1}{x^{2}}-x^{2}-\frac{1}{3}\right) d x$$

Step-by-Step Solution

Answer

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

Step 1: Decompose the Integral

The given integral is \( \int \left( \frac{1}{x^2} - x^2 - \frac{1}{3} \right) \, dx \). We can decompose this into three separate integrals: \( \int \frac{1}{x^2} \, dx \), \( \int -x^2 \, dx \), and \( \int -\frac{1}{3} \, dx \).

Step 2: Integrate Each Term

Let's find the integral of each term separately: 1. For \( \int \frac{1}{x^2} \, dx \): This is equivalent to \( \int x^{-2} \, dx \). The antiderivative of \( x^{-n} \) is \( -\frac{1}{x} \), so \( -\frac{1}{x} + C_1 \).2. For \( \int -x^2 \, dx \): The antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} \), so this becomes \( -\frac{x^3}{3} + C_2 \).3. For \( \int -\frac{1}{3} \, dx \): This is a constant, so its integral is \( -\frac{1}{3}x + C_3 \).

Step 3: Combine the Results

Now, combine all the antiderivatives:\(-\frac{1}{x} - \frac{x^3}{3} - \frac{1}{3}x + C\).Here, \(C = C_1 + C_2 + C_3\) is the constant of integration, accounting for all individual constants.

Step 4: Verify by Differentiation

Differentiate the combined function to check for correctness:Take the derivative of \(-\frac{1}{x} - \frac{x^3}{3} - \frac{1}{3}x + C\):- Derivative of \(-\frac{1}{x}\) is \(\frac{1}{x^2}\).- Derivative of \(-\frac{x^3}{3}\) is \(-x^2\).- Derivative of \(-\frac{1}{3}x\) is \(-\frac{1}{3}\).Combining these, we get back the original function: \(\frac{1}{x^2} - x^2 - \frac{1}{3}\). This confirms our solution is correct.

Key Concepts

Understanding Antiderivatives

An antiderivative is a function whose derivative gives back the original function. In other words, it's the reverse process of differentiation. Finding an antiderivative means computing an indefinite integral. Every function can have infinitely many antiderivatives, differing by a constant called the constant of integration. This constant appears because the derivative of a constant is zero, and thus constant terms vanish during differentiation.

To find an antiderivative, you attempt to determine the original function based on its derivative. In the original exercise, we were tasked to determine the most general antiderivative of the function \( \frac{1}{x^2} - x^2 - \frac{1}{3} \). This involves identifying a function whose derivative results in the given expression. Once found, we verified its correctness by differentiating it and comparing it back to the original expression.

How Differentiation Confirms Calculations

Differentiation is used to confirm that the antiderivative we found is correct. In simpler terms, differentiation checks our work. Once we compute an antiderivative or indefinite integral, we differentiate it to ensure we return to the original function.

This process works because differentiation and integration are inverse operations. In our exercise, after computing the antiderivative of \( \int \left( \frac{1}{x^2} - x^2 - \frac{1}{3} \right) \, dx \), we differentiated the resulting expression to verify it. Calculating the derivative as surrounding each term, such as \( -\frac{1}{x} \) became \( \frac{1}{x^2} \) upon differentiation, confirmed our antiderivative was calculated correctly. This backward check is essential for ensuring accuracy.

Exploring Integration Techniques

Integration techniques help solve integrals more efficiently, especially when dealing with complex functions. These techniques include decomposition, substitution, integration by parts, and others. Each method is suited to specific types of integrands, simplifying the integration process to make manual calculations feasible.

For the exercise at hand, we used the decomposition method. The integral \( \int \left( \frac{1}{x^2} - x^2 - \frac{1}{3} \right) \, dx \) was split into three simpler integrals: \( \int \frac{1}{x^2} \, dx \), \( \int -x^2 \, dx \), and \( \int -\frac{1}{3} \, dx \). Each was solved individually, as each had straightforward antiderivatives. After computing each, they were recombined to form the most general solution for the original integral. Breaking down integrals into smaller, manageable parts exemplifies the power of decomposition in simplifying complex integral expressions.