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Chapter 4: Problem 31

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

Short Answer

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

Step by step solution

01

Expand the Integrand

Rewrite the integrand by expanding the expression inside the integral: \[\int 2x(1-x^{-3})\, dx = \int (2x - 2x\cdot x^{-3})\, dx = \int (2x - 2x^{-2})\, dx\]
02

Find the Antiderivative of Each Term

Integrate each term separately:- For \(2x\), apply the power rule: \[ \int 2x\, dx = 2 \cdot \frac{x^{1+1}}{1+1} = x^2 \]- For \(2x^{-2}\), apply the power rule:\[ \int -2x^{-2}\, dx = -2 \cdot \frac{x^{-2+1}}{-2+1} = 2x^{-1} = \frac{2}{x} \]
03

Combine the Antiderivatives

Add the antiderivatives together and include the constant of integration \(C\): \[x^2 + \frac{2}{x} + C\]
04

Check the Solution by Differentiation

Differentiate the result to check if it matches the original integrand:- Differentiate \(x^2\):\[ \frac{d}{dx}(x^2) = 2x \]- Differentiate \(\frac{2}{x}\):\[ \frac{d}{dx}\left(\frac{2}{x}\right) = \frac{d}{dx}(2x^{-1}) = -2x^{-2} \]Combine these derivatives:\[ 2x - 2x^{-2} \]This matches the original integrand, confirming our solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Indefinite Integrals
Indefinite integrals, also called antiderivatives, are a foundational concept in calculus. When we talk about an indefinite integral, we mean finding the original function whose derivative gives the integrand (the function being integrated). Essentially, we are performing the reverse of differentiation.

To denote the indefinite integral, we use the integral symbol (鈭�) followed by the function and then the symbol 'dx,' indicating the variable of integration. The result of an indefinite integral, unlike a definite integral, includes a constant of integration, represented by 'C.' This constant accounts for all the possible vertical shifts of the original function that share the same derivative.

  • The process involves integrating each term in the expression individually.
  • Linear properties of integration allow us to do this term by term.
For example, if you're given the integral of the form \(\int (f(x) + g(x)) \, dx\), you can use linearity to separate it into \(\int f(x) \, dx + \int g(x) \, dx\). This separation simplifies the integration process.
Power Rule of Integration
The power rule of integration is one of the most commonly used techniques for finding antiderivatives. This rule is quite simple and is applied separately to each term in a polynomial expression.
The power rule states that the integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\), plus a constant \(C\).

  • This rule applies as long as \(n eq -1\).
  • It's important to adjust the exponent and the coefficient properly when applying this rule.
However, when \(n = -1\), the power rule does not apply. Instead, the antiderivative is the natural logarithm function. For \(x^{-1}\), the result is \(\ln |x|\).

This rule is very efficient in breaking down expressions like \(2x - 2x^{-2}\) as seen in the exercise. The application of the power rule simplifies the task of finding the antiderivative, making it a go-to technique in calculus. In the example provided:
  • For \(2x\), it becomes \(x^2\).
  • For \(2x^{-2}\), it simplifies to \(\frac{2}{x}\).
Differentiation Check
After finding an indefinite integral, it is crucial to verify that our solution is correct. This is done through a differentiation check. The differentiation check involves differentiating the antiderivative and making sure that you arrive back at the original integrand.

  • This step ensures that no errors were made during integration.
  • It provides a point of reassurance, confirming the accuracy of the integral calculation.
In our example, after integrating to get \(x^2 + \frac{2}{x} + C\), differentiating each term should bring back the original expression inside the integral.

When you differentiate \(x^2\), you get \(2x\). Similarly, when differentiating \(\frac{2}{x}\), you arrive at \(-2x^{-2}\). When combined, these results equal the original integrand \(2x - 2x^{-2}\), confirming the integration was done correctly. Hence, this differentiation check serves as not only a verification tool but also as a confidence booster for the solution obtained.

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Most popular questions from this chapter

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