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Chapter 7: Problem 6

Describe a first step in integrating \(\int \frac{x^{10}-2 x^{4}+10 x^{2}+1}{3 x^{3}} d x\).

Short Answer

Expert verified
Question: Find the integral of the rational function \(\int \frac{x^{10}-2 x^{4}+10 x^{2}+1}{3x^{3}} d x\). Answer: The integral of the given rational function is \(\frac{x^{8}}{24} + \frac{x^{2}}{3} + \frac{10}{3} \ln(|x|) - \frac{1}{6x^{2}} + C\).

Step by step solution

01

Break down the numerator

Divide each term in the numerator by the denominator to get: \(\int \frac{x^{10}}{3x^{3}} d x - \int \frac{2 x^{4}}{3x^{3}} d x + \int \frac{10 x^{2}}{3x^{3}} d x + \int \frac{1}{3x^{3}} d x\)
02

Simplify each term

Simplify the terms inside the integrals to get: \(\int \frac{1}{3}x^{7} d x - \int \frac{2}{3}x d x + \int \frac{10}{3x} d x + \int \frac{1}{3x^{3}} d x\)
03

Integrate each term

Integrate each term separately: 1. \(\int \frac{1}{3}x^{7} d x = \frac{1}{3} \frac{x^{8}}{8} + C_1 = \frac{x^{8}}{24} + C_1\) 2. \(\int \frac{2}{3}x d x = \frac{2}{3} \frac{x^{2}}{2} + C_2 = \frac{x^{2}}{3} + C_2\) 3. \(\int \frac{10}{3x} d x = \frac{10}{3} \ln(|x|) + C_3\) 4. \(\int \frac{1}{3x^{3}} d x = \frac{1}{3} \frac{-1}{2x^{2}} + C_4 = -\frac{1}{6x^{2}} + C_4\)
04

Combine the integrated terms

Combine all the integrated terms and their constants into a single expression: \(\frac{x^{8}}{24} + \frac{x^{2}}{3} + \frac{10}{3} \ln(|x|) - \frac{1}{6x^{2}} + C\)

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Key Concepts

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

Polynomial Division
When you encounter a fraction where the numerator is a polynomial of higher degree than the denominator, polynomial division is a useful initial step. This technique is essential before simplifying integrals, making future integration steps easier. In our exercise, the division of each term in the numerator by the denominator serves to break down the complex expression into simpler, individual terms.
  • Step 1: Look at each term in the numerator separately. Divide each term by the denominator.
  • Example: For the term \(x^{10}\), dividing by \(3x^3\) gives \(\frac{x^{10}}{3x^3} = \frac{1}{3}x^{7}\).
This division makes each part of the integral more manageable and ready for subsequent steps, such as simplification or direct integration.
Simplification of Integrals
Simplifying integrals is crucial in the integration process, as it reduces the complexity and facilitates easier integration. After polynomial division, you might end up with simpler terms that can be easily integrated or even terms that directly correspond to basic integral formulas.
  • Step 2: Each term from the division can often be reduced further. Simplify terms as needed.
  • Example: Rather than integrating \(\frac{x^4}{x^3}\) directly, simplify it to \(x\) before integrating.
Simplification makes your work more straightforward and helps avoid potential errors in integration by only working with the most simplified version of each term.
Integration of Rational Functions
Rational functions, which are quotients of polynomials, often require special handling during integration. Once simplified, many rational functions may be ready to integrate. However, sometimes additional techniques like partial fraction decomposition may be needed if they cannot be integrated directly.
  • Direct integration: Some simplified rational functions can be integrated directly using known integration formulas.
  • Example: \(\int \frac{10}{3x}\) uses the natural log rule, resulting in \(\frac{10}{3}\ln(|x|)\).
Using a mix of direct integration and additional techniques, like logarithmic integration, helps tackle these functions efficiently.
Indefinite Integrals
An indefinite integral, also known as an antiderivative, represents a family of functions whose derivative is the given function. When calculating indefinite integrals, each term is integrated separately, often from its simplified form, to provide a function plus an arbitrary constant \(C\).
  • Step 3: Solve the integral for each term.
  • Example: The integral of \(\frac{1}{3}x^7\) becomes \(\frac{x^8}{24} + C\).
Finally, combine these terms and their respective constants into one expression. The arbitrary constant accounts for any vertical shifts in the antiderivative, emphasizing why indefinite integrals are expressed as a family of functions.

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