Problem 6 - 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. \(\begin{array}{llll}{\text { a. }-\frac{2}{x^{3}}} & {\text { b. } \frac{1}{2 x^{3}}} & {\text { c. } x^{3}-\frac{1}{x^{3}}}\end{array}\)

Step-by-Step Solution

Answer

Antiderivatives: a. \( \frac{1}{x^{2}} + C \), b. \( -\frac{1}{4x^{2}} + C \), c. \( \frac{x^{4}}{4} + \frac{1}{2x^{2}} + C \).

Step 1: Finding the antiderivative of (a)

The function given is \( -\frac{2}{x^{3}} \). We can rewrite this as \( -2x^{-3} \). To find the antiderivative, we apply the power rule for integration, which states that the antiderivative of \( x^{n} \) is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. Therefore, the antiderivative is $$\frac{-2x^{-2}}{-2} + C = x^{-2} + C$$, or \( \frac{1}{x^{2}} + C \).

Step 2: Verification of (a) by differentiation

To verify, we differentiate the antiderivative \( x^{-2} + C \). The derivative of \( x^{-2} \) is \( -2x^{-3} \). As expected, this derivative matches the original function \( -\frac{2}{x^{3}} \), confirming our antiderivative is correct.

Step 3: Finding the antiderivative of (b)

The function given is \( \frac{1}{2x^{3}} \), which can be rewritten as \( \frac{1}{2}x^{-3} \). Applying the power rule, the antiderivative is \( \frac{1}{2}\cdot\frac{x^{-2}}{-2} + C = -\frac{1}{4}x^{-2} + C \), which simplifies to \( -\frac{1}{4x^{2}} + C \).

Step 4: Verification of (b) by differentiation

Differentiate the antiderivative \( -\frac{1}{4}x^{-2} + C \). The derivative calculation gives us \( -\frac{1}{4} \cdot (-2)x^{-3} = \frac{1}{2}x^{-3} \). This matches the original function, confirming our solution is correct.

Step 5: Finding the antiderivative of (c)

For the function \( x^{3} - \frac{1}{x^{3}} \), separate terms and find their antiderivatives individually. The antiderivative of \( x^{3} \) is \( \frac{x^{4}}{4} \). For \( -\frac{1}{x^{3}} = -x^{-3} \), the antiderivative is \( \frac{x^{-2}}{-2} = \frac{1}{2}x^{-2} \). Combining these, the function's antiderivative is \( \frac{x^{4}}{4} + \frac{1}{2x^{2}} + C \).

Step 6: Verification of (c) by differentiation

Differentiate the antiderivative \( \frac{x^{4}}{4} + \frac{1}{2x^{2}} + C \). The derivative of \( \frac{x^{4}}{4} \) is \( x^{3} \), and the derivative of \( \frac{1}{2x^{2}} \) is \( -\frac{1}{x^{3}} \). Combining them, the result is \( x^{3} - \frac{1}{x^{3}} \), matching the original function and confirming the antiderivative is correct.

Key Concepts

Power Rule for Integration

Integration is like the reverse process of differentiation. When we integrate a function, we are essentially looking for its antiderivative, a function whose derivative is the given function. One of the most common techniques used in integration is the **Power Rule for Integration**.

The Power Rule is a straightforward formula that helps us find antiderivatives quickly. If you have a function in the form of \( x^n \), you can find its antiderivative by applying the formula:

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

where \( C \) is the constant of integration, which accounts for any constant value that would disappear during differentiation. This rule applies when \( n eq -1 \).

For example, in the first problem, the function \(-\frac{2}{x^3}\) was rewritten as \(-2x^{-3}\). By using the Power Rule, we determined the antiderivative to be:

• \[ \int -2x^{-3} \, dx = x^{-2} + C \]

This solution shows how powerful and simple the Power Rule can be when applied correctly.

Differentiation

To ensure that we have found the correct antiderivative, it is crucial to use differentiation. Differentiation is the process of finding the derivative of a function, which measures how a function changes as its input changes.

When you differentiate the antiderivative, and if it results in the original function, you can be confident that you have found the correct solution. Differentiation serves as a verification step in these types of integration problems.

Consider the antiderivative \( x^{-2} + C \) for the original function \(-\frac{2}{x^3}\). Differentiating \( x^{-2} \) we apply the power rule for differentiation, which states that the derivative of \( x^n \) is \( nx^{n-1} \):

• \( \frac{d}{dx}(x^{-2}) = -2x^{-3} \)

This matches the original function \(-\frac{2}{x^3}\), confirming that our antiderivative is indeed correct. This reverse-check confirms the accuracy of our integration process.

Constant of Integration

Whenever we find an antiderivative, we introduce the constant of integration, represented by the symbol \( C \). This constant is crucial because when we differentiate a constant, it disappears (the derivative of a constant is zero).

This means that any number could have been added to our antiderivative before differentiation without affecting the result of the differentiation process. Therefore, when writing the solution to an indefinite integral, it's important to include the \( C \) to account for all possible constants that could have existed in the original function.

In the example, when finding the antiderivative of \( -\frac{2}{x^3} \), we have:

• \[ \int -2x^{-3} \, dx = x^{-2} + C \]

Here, \( C \) represents any constant, reminding us that there isn't a single antiderivative, but rather a family of functions differing by a constant. Including \( C \) ensures our solution reflects this infinite possibility of originals, leading to a holistic understanding of integration.