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Chapter 7: Problem 4

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

Short Answer

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Question: Determine the integral of the rational function \(f(x) = \frac{x^3 - 2x + 4}{x-1}\). Answer: \(\int f(x) dx = \frac{x^3}{3} + \frac{x^2}{2} - 2x + 3\ln|x-1| + C\)

Step by step solution

01

Polynomial Long Division

Divide the numerator, \(x^3 - 2x + 4\), by the denominator, \(x-1\): $$ \require{cancel} \begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} & +x & -2 \\ \cline{1-1} x - 1 & x^3& - 3x^2& \cancel{+x^2} &+4\\ \cline{1-2} & -x^3 & +x^2 \\ \cline{1-2} & 0 & -2x^2 & +x \\ \cline{2-4} & & 0 & +x & -2 \\ \cline{2-3} & & & -x &+1 \\ \cline{3-5} \end{array} $$ So, \(\frac{x^3 - 2x + 4}{x-1} = x^2 + x - 2 + \frac{3}{x-1}\). Now we proceed to integrate the resulting expression.
02

Integrate the expression

Integrate each term separately: $$\int \frac{x^{3}-2 x+4}{x-1} d x = \int \left(x^2 + x - 2 + \frac{3}{x-1}\right) dx.$$ $$\int (x^2 + x - 2)dx + \int \frac{3}{x-1}dx = \frac{x^3}{3} + \frac{x^2}{2} - 2x + 3\ln|x-1|+C .$$ The integral is equal to: $$\int \frac{x^{3}-2 x+4}{x-1} d x = \frac{x^3}{3} + \frac{x^2}{2} - 2x + 3\ln|x-1| + C$$

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Key Concepts

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

Polynomial Long Division
Polynomial long division is similar to numerical long division but is used on polynomial expressions. It helps break down complex rational expressions into simpler ones by dividing the numerator by the denominator.

In the context of our integral,
  • We start with the numerator: a polynomial of degree 3, which is \(x^3 - 2x + 4\).
  • We divide it by the polynomial \(x - 1\).
We must subtract the product from the original polynomial, step by step, across all terms. Doing this accurately reveals a simpler expression:
  • \(x^2 + x - 2\)
  • and a remainder \(\frac{3}{x-1}\).
This division simplifies the integration process by turning a complex rational function into a sum of simpler polynomials and a remainder term. With practice, this method becomes straightforward and a valuable tool in calculus.
Definite Integration
Definite integration is a process to find the exact numerical value of an integral over a specified interval. Our exercise involves indefinite integration, which can later support definite integration by providing the antiderivative. Calculating definite integrals
  • determines the total accumulated area under a curve between two points on the x-axis.
  • it is expressed as \( \int_a^b f(x)\,dx \), where \(a\) and \(b\) are the limits of integration.
For tasks like these, understanding the indefinite integral can be foundational. After computing the indefinite form, you can apply the limits to find a specific area. Remember, definite integration provides insights into values without arbitrary constants, grounding its result in real numbers.
Antiderivatives
Antiderivatives, also called indefinite integrals, are functions that reverse differentiation. When integrating, the goal is to find the function whose derivative matches the given function.

In the provided exercise, the antiderivative of the simplified polynomial expression \(x^2 + x - 2\) and \(\frac{3}{x-1}\) needs to be identified.
  • For \(x^2\), an antiderivative is \(\frac{x^3}{3}\).
  • For \(x\), it is \(\frac{x^2}{2}\).
  • For \(-2\), \(-2x\).
  • For \(\frac{3}{x-1}\), it's \(3\ln|x-1|\).
In the final expression \(\frac{x^3}{3} + \frac{x^2}{2} - 2x + 3\ln|x-1| + C\), each antiderivative reflects a term from the original polynomial breakdown. The constant \(C\) represents the "general" nature of indefinite integration and its potential to shift, accommodate, and align functions to specific scenarios or constraints.

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Most popular questions from this chapter

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