Problem 31 - Thomas Calculus

Question

In Exercises \(17-56,\) 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 antiderivative is \(x^2 + \frac{2}{x} + C\).

Step 1: Expand the integrand

First, distribute the term \(2x\) inside the parentheses: \[2x(1 - x^{-3}) = 2x - 2x^{1 - 3} = 2x - 2x^{-2}\]

Step 2: Antidifferentiate each term separately

Find the antiderivative for each term:For the term \(2x\):\[\int 2x \, dx = 2 \cdot \frac{x^2}{2} = x^2\]For the term \(-2x^{-2}\):\[\int -2x^{-2} \, dx = -2 \left(-\frac{x^{-1}}{1}\right) = 2x^{-1}\]Combine both results:\[\int (2x - 2x^{-2}) \, dx = x^2 + 2x^{-1} + C\]

Step 3: Verify the result by differentiation

Differentiate the obtained expression to ensure it matches the original integrand:\[\frac{d}{dx}(x^2 + 2x^{-1} + C) = 2x - 2x^{-2}\]This matches the original integrand expression, confirming the integration was performed correctly.

Key Concepts

Integration

Integration is a cornerstone concept in calculus that deals with finding the whole from knowing the parts. Unlike differentiation, which breaks functions down into their rate of change, integration aggregates values to find total amounts. This concept is integral (pun intended!) in calculating areas, volumes, and other quantities where accumulation is involved.

• In the solution provided, the integration process takes the modified expression from step 1 through a systematic process to find its antiderivative.
• The notation used in integration involves the integral symbol \(\int\), which suggests adding up values over a range.
• In this case, after expanding the expression to have terms like \(2x\) and \(-2x^{-2}\), these are integrated separately to find the result.

It's fascinating how the static nature of integrals contrasts with the dynamic nature of derivatives, yet both are crucial parts of the calculus universe.

Differentiation

Differentiation, as highlighted in this exercise, serves as the checking tool for verifying our integration result. It is the process of finding the rate at which a function is changing at any given point. Since integration and differentiation are inverse operations, you can use differentiation to confirm your integration results.

• In the solution, after calculating the indefinite integral, the expression \(x^2 + 2x^{-1} + C\) is differentiated.
• The differentiation yields \(2x - 2x^{-2}\), which matches the original integrand, confirming the correctness of the integration process.
• This verification step is crucial because even minor errors in integration can lead to incorrect results.

Understanding differentiation deeply enhances the ability to see how functions react to change, and it pairs perfectly with integration to form the bedrock of calculus.

Calculus

Calculus, combining differentiation and integration, extends beyond just mathematical exercises into practical applications in various fields such as physics, engineering, economics, and beyond. It provides the tools necessary for modeling and understanding dynamic systems.

• One of the core themes in calculus exercises like this one is finding connections between change (differentiation) and accumulation (integration).
• The exercise demonstrates this duality by requesting the indefinite integral (an accumulation) and asking for validation via differentiation.
• Through this, calculus reveals how seemingly static mathematical expressions can describe dynamic real-world problems.

Though challenging, mastering calculus opens a universe of problem-solving strategies and insights that are unparalleled in understanding the world.

Indefinite Integral

The concept of an indefinite integral is all about finding the most general form of an antiderivative for a given function. Unlike a definite integral, which calculates a specific numerical value representing area under a curve, an indefinite integral includes a constant of integration \(C\) because it represents a family of functions.

• In the provided exercise, the indefinite integral \(\int (2x - 2x^{-2}) \, dx\) leads to the family of functions expressed as \(x^2 + 2x^{-1} + C\).
• The \(C\) is crucial as it accounts for any constant that could have been in the original function, which would disappear when differentiating.
• This fundamental idea in calculus emphasizes the fluidity and flexibility of functions and their antiderivatives.

Indefinite integrals thus play a crucial role in calculus, highlighting the uncertainties and infinite possibilities that stem from reverse-engineering derivatives.