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Chapter 5: Problem 38

Find each indefinite integral. \(\int \frac{x^{2}-1}{x-1} d x\)

Short Answer

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

Step by step solution

01

Simplify the Expression

First, simplify the expression inside the integral. Consider the polynomial division. Divide \(x^2 - 1\) by \(x - 1\). Perform the division to obtain \(x + 1\) as the result. Thus, \(\frac{x^2 - 1}{x - 1} = x + 1\).
02

Rewrite the Integral

Now that the expression is simplified to \(x + 1\), rewrite the integral accordingly: \[\int \frac{x^2 - 1}{x - 1} \, dx = \int (x + 1) \, dx.\]
03

Integrate Term-wise

Integrate each term separately. The integral of \(x\) is \(\frac{x^2}{2}\), and the integral of \(1\) is \(x\). Write down the result of integration:\[ \int (x + 1) \, dx = \frac{x^2}{2} + x + C,\] where \(C\) is the constant of integration.
04

Final Answer

Combine the results from the integration to write out the final answer:\[ \int \frac{x^2 - 1}{x - 1} \, dx = \frac{x^2}{2} + x + C.\]

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Key Concepts

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

Polynomial Division: Simplifying the Expression
Polynomial division is a technique used to simplify expressions involving polynomials, much like how division simplifies regular numbers. In the exercise, we are dealing with the expression \( \frac{x^2 - 1}{x - 1} \). This is a division of one polynomial by another.

The key steps in polynomial division are:
  • Divide the first term of the numerator by the first term of the denominator.
  • Multiply the entire divisor by this result and subtract it from the original numerator.
  • Repeat the process with the new polynomial that results from the subtraction.
In this exercise, dividing \( x^2 \) by \( x \) gives \( x \). Multiplying the entire divisor \((x-1)\) by \( x \) results in \(x^2 - x\), which when subtracted from \(x^2 - 1\), leaves us with \(x + 1\).

Our polynomial division simplifies the expression to \(x + 1\), making the subsequent integration straightforward.
Integration Techniques: Finding the Indefinite Integral
Integration is a core component of calculus used to find the cumulative area under curves. In integration techniques, breaking down complex expressions into simpler ones is a key strategy, as seen in our exercise.

After simplifying the integral expression to \(\int (x + 1) \, dx\), we use term-wise integration. Each term in the expression is tackled individually. Some basic rules of integration applied here include:
  • The integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\) for any constant \(n\).
  • The integral of a constant term, such as \(1\), is the term multiplied by \(x\).
For \(x\), the integral becomes \(\frac{x^2}{2}\), as the power of \(x\) increases by one, and the result is divided by the new exponent. For the constant \(1\), the integral is simply \(x\). Together, these integration techniques provide \(\frac{x^2}{2} + x\).
Constant of Integration: Accounting for Generality
In indefinite integrals, you often encounter the symbol \(C\). This represents the constant of integration. But what exactly does it mean?

While evaluating definite integrals yields a specific numerical answer, indefinite integrals describe a family of functions. An indefinite integral lacks specific limits, which means there are countless functions that differ only by a constant amount that can be added.
  • The result of an indefinite integral is a general solution that can be written as a function plus an arbitrary constant \(C\).
  • This constant \(C\) accounts for any constant value that would have been differentiated to zero in the original function.
For our exercise \(\int (x + 1) \, dx\), the final answer becomes \(\frac{x^2}{2} + x + C\). Including \(C\) ensures that we represent all possible antiderivatives of the function.

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