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

Step-by-Step Solution

Answer

The most general antiderivative is \(x^2 + \frac{2}{x} + C\).

Step 1: Simplify the Integrand

The given integral is \( \int 2x(1-x^{-3})\,dx \). First, simplify the integrand by distributing the \(2x\) inside the parentheses: \(2x \cdot 1 - 2x \cdot x^{-3} = 2x - 2x^{1-3} = 2x - 2x^{-2}\). Now the integral is: \(\int (2x - 2x^{-2})\,dx\).

Step 2: Integrate Term by Term

Integrate each term separately. - For \(2x\): Use the power rule, \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\). Here \(n=1\), so: \(\int 2x\,dx = 2 \cdot \frac{x^{1+1}}{1+1} = x^2\).- For \(-2x^{-2}\): Again using the power rule. Here \(n = -2\), so: \(\int -2x^{-2}\,dx = -2 \cdot \frac{x^{-2+1}}{-2+1} = 2x^{-1} = 2 \cdot \frac{1}{x}\).Combine these results: \(x^2 + 2x^{-1} = x^2 + \frac{2}{x}\).

Step 3: Add the Constant of Integration

Since we are finding the antiderivative, include the constant of integration \(C\). The most general antiderivative of the given function is: \(x^2 + \frac{2}{x} + C\).

Step 4: Verify by Differentiating

Differentiate the antiderivative to check your work.- Differentiate \(x^2\): \(\frac{d}{dx}(x^2) = 2x\).- Differentiate \(\frac{2}{x}\): \(\frac{d}{dx}(\frac{2}{x}) = -2x^{-2}\).- \(\frac{d}{dx}(C) = 0\).Combine these results: \(2x - 2x^{-2}\) matches the original integrand \(2x - 2x^{-2}\), confirming the integration was performed correctly.

Key Concepts

Power Rule

The Power Rule is a fundamental tool in calculus that is immensely helpful when finding integrals, specifically indefinite integrals. It is used to integrate expressions in the form of powers of x. When dealing with an integral of the form:

• \( \int x^n \, dx \)

the Power Rule simplifies the process by allowing us to compute the antiderivative as:

• \( \frac{x^{n+1}}{n+1} + C \)

Here, \( n \) is any real number except \(-1\) because it leads to a division by zero error, and \( C \) represents the constant of integration.
In the solution, we apply the Power Rule to each term of the integrand separately. For example, consider the term \( 2x \). It can be rewritten as \( 2x^1 \), and using the Power Rule gives us the antiderivative \( x^2 \). The technique remains similar for negative exponents, such as \(-2x^{-2}\), which integrate to \( 2 \cdot \frac{1}{x} \).
By extending this simple rule, calculus students can tackle more complex integrals by breaking them into smaller, more manageable parts. This rule forms the backbone of integration methods, validating its indispensable role in learning calculus.

Antiderivative

An antiderivative is essentially the reverse process of differentiation. While differentiation tells us how functions change, antiderivatives point back to the original function whose derivative yields the given function. This is why it is often referred to as finding the "indefinite integral," noted by the integral symbol \( \int \).
In this context, if you have a function \( f(x) \) and its derivative is \( f'(x) \), the function \( f(x) \) is the antiderivative of \( f'(x) \). Unlike definite integrals, which provide a numerical result, indefinite integrals offer a family of functions, all differing by a constant.
Returning to our original exercise, the antiderivative of the function \( 2x(1-x^{-3}) \) is found by integrating each portion separately, and combining results to form a single expression. This cumulative function, including \( x^2 + \frac{2}{x} + C \), represents the most generalized form of the function that provides the original derivative if differentiated.
Understanding this concept is vital, as antiderivatives open the door to solving more complex areas of subfields within mathematics and applied sciences.

Constant of Integration

The Constant of Integration is a critical component in the realm of indefinite integrals. Represented by the symbol \( C \), this constant accounts for the fact that there could be an infinite number of functions that all have the same derivative.
Whenever an indefinite integral is evaluated, it could result in a family of antiderivatives. Each member of this family differs only by a constant value. This constant is absent in a definite integral or when taking derivatives, as differentiation will eliminate any constants.
In the step-by-step solution, once the terms of the indefinite integral are integrated, the constant \( C \) is added to complete the expression as \( x^2 + \frac{2}{x} + C \). This addition is a reminder that without specific limits (i.e., boundaries), any constant added would still result in the same derivative being obtained.
By grasping the purpose and necessity of the constant of integration, students can better understand the nature of indefinite integrals, ensuring a correct and complete solution.