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Chapter 5: Problem 12

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

Short Answer

Expert verified
\( \int \left( x - \frac{1}{x} \right)^2 \, dx = \frac{x^3}{3} - 2x - \frac{1}{x} + C \)

Step by step solution

01

Expand the expression

First, expand the expression inside the integral. \[ (x - \frac{1}{x})^{2} = x^2 - 2x \cdot \frac{1}{x} + (\frac{1}{x})^2 \] Further simplify this:\[ x^2 - 2 + \frac{1}{x^2} \]
02

Integrate each term

Now integrate each term separately: \[ \int x^2 \, dx, \quad \int -2 \, dx, \quad \int \frac{1}{x^2} \, dx \] 1. **\( \int x^2 \, dx \):** \[ \frac{x^3}{3} + C_1 \]2. **\( \int -2 \, dx \):** \[ -2x + C_2 \] 3. **\( \int \frac{1}{x^2} \, dx \):** \[ -\frac{1}{x} + C_3 \] Combine the results to get the total integral.
03

Write out the final result

Combine the integrated terms to find the final answer:\[ \frac{x^3}{3} - 2x - \frac{1}{x} + C \]\( C \) represents the constant of integration, which is the sum: \( C = C_1 + C_2 + C_3 \).

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Key Concepts

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

Definite Integrals
Definite integrals allow us to find the accumulated quantity over an interval. Unlike indefinite integrals, definite integrals provide a numerical result because they have specific limits on their integration range. In practice, they are used to calculate areas under curves, among other applications.
To evaluate a definite integral, you typically:
  • Find the indefinite integral (antiderivative) of the function.
  • Apply the Fundamental Theorem of Calculus by evaluating this antiderivative at the upper and lower bounds of the interval.
  • Subtract the lower bound value from the upper bound value to get the result.
Definite integrals help bridge calculus and real-world measurements by turning equations into tangible data like area and volume.
Polynomial Integration
Polynomial integration involves integrating functions expressed as polynomials, which is generally straightforward. A polynomial is a sum of terms, each consisting of a constant multiplied by a variable raised to a power.For instance, in this problem, we integrated terms like \(x^2\) and constants like -2.
  • To integrate a term like \(x^n\), use the rule: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \(C\) is the constant of integration.
  • This rule applies to each term separately, then sum the results to achieve the final integral.
Even when thematically simple, polynomial integration builds the foundation for more complex calculus operations, ensuring you grasp how basic calculus shapes more advanced processes.
Antiderivatives
Antiderivatives, also known as indefinite integrals, are the reverse of derivatives. Finding an antiderivative involves determining a function whose derivative is the given function.
A key aspect of antiderivatives is the constant of integration, represented as \(C\). Since differentiation erases constant values, when finding an antiderivative, there's always an unknown constant added back.
  • For example, when finding an antiderivative of \(x^2\), you'll find \(\frac{x^3}{3} + C\).
  • Antiderivatives require understanding derivative rules backward, like knowing that the inverse of deriving power functions will increase the power by one and divide by the new power.
Mastering antiderivatives allows solving differential equations and finding areas under curves, essential skills in calculus.

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Most popular questions from this chapter

(a) Find a number \(z\) that satisfies the conclusion of the mean value theorem (5.28) for the given integral \(\int_{a}^{b} f(x) d x .\) (b) Find the average value of \(f\) on \([a, b]\). $$ \int_{-2}^{0} \sqrt[3]{x+1} d x $$

Evaluate. $$ \int_{1}^{1} 3 x^{2} \sqrt{x^{3}+x} d x $$

Find the derivative without integrating. $$ D_{x} \int_{0}^{x} \frac{1}{t+1} d t $$

Evaluate. $$ \int 100 d x $$

Evaluate \(\int_{0}^{10} \sqrt{1+x^{4}} d x\) by using (a) the trapezoidal rule, with \(n=5\) (b) Simpson's rule, with \(n=8\) (Use approximations to four decimal places for \(f\left(x_{k}\right)\), and round off answers to two decimal places.)
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