/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Find \(\int \frac{x^{18}-1}{x^{3... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 25

Find \(\int \frac{x^{18}-1}{x^{3}-1} d x\)

Short Answer

Expert verified
Integrate to get \(\frac{x^{16}}{16} + \frac{x^{13}}{13} + \frac{x^{10}}{10} + \frac{x^{7}}{7} + \frac{x^{4}}{4} + x + C\).

Step by step solution

01

Polynomial Division

The integrand can be simplified by performing polynomial long division of \(x^{18} - 1\) by \(x^3 - 1\). First, understand that \(x^{18} - 1 = (x^3 - 1)((x^3)^5 + (x^3)^4 + (x^3)^3 + (x^3)^2 + x^3 + 1)\). Thus, upon division we have:\[x^{18} - 1 = Q(x)(x^3 - 1) + R(x)\]where \(Q(x) = x^{15} + x^{12} + x^9 + x^6 + x^3 + 1\) and \(R(x) = 0\). Therefore, the integrand becomes \(Q(x)\):\[\int rac{x^{18} - 1}{x^{3} - 1} dx = \int (x^{15} + x^{12} + x^9 + x^6 + x^3 + 1) dx.\]
02

Perform Integration

Now, integrate each term separately. We have:\[\begin{align*}\int (x^{15} + x^{12} + x^9 + x^6 + x^3 + 1) dx &= \int x^{15} dx + \int x^{12} dx + \int x^9 dx + \int x^6 dx + \int x^3 dx + \int 1 dx \&= \frac{x^{16}}{16} + \frac{x^{13}}{13} + \frac{x^{10}}{10} + \frac{x^{7}}{7} + \frac{x^{4}}{4} + x + C\end{align*}\]where \(C\) is the constant of integration.

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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 a method used to divide one polynomial by another. It's similar to the long division you use with numbers, but here you deal with variables and their exponents. In this exercise, we divide \(x^{18} - 1\) by \(x^3 - 1\). This is necessary because our integrand can be quite complex without it.

  • First, recognize that the polynomial \(x^{18} - 1\) can be rewritten as \((x^3 - 1)((x^3)^5 + (x^3)^4 + (x^3)^3 + (x^3)^2 + x^3 + 1)\).
  • Then, perform the division to reveal how many complete times \(x^3 - 1\) fits into the dividend without leaving a remainder.
  • Once completed, you find \(Q(x)\) which is \(x^{15} + x^{12} + x^9 + x^6 + x^3 + 1\), and \(R(x) = 0\), indicating there's no remainder.
This simplification helps in breaking down the integrand into individual polynomial terms, making it easier to integrate.
Integration Techniques
Integration is the process of finding a function whose derivative is the given function, known as the antiderivative. With polynomial functions, integration is straightforward as each term in the polynomial is integrated separately. Let's apply this to the polynomial we obtained from the division stage:

  • The integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\). This is the basic rule applied repetitively here.
  • Integrating our polynomial \(x^{15} + x^{12} + x^9 + x^6 + x^3 + 1\) term by term gives us:
    • \(\int x^{15} dx = \frac{x^{16}}{16}\)
    • \(\int x^{12} dx = \frac{x^{13}}{13}\)
    • \(\int x^9 dx = \frac{x^{10}}{10}\)
    • \(\int x^6 dx = \frac{x^7}{7}\)
    • \(\int x^3 dx = \frac{x^4}{4}\)
    • \(\int 1 dx = x\)
  • Adding all these results gives us the antiderivative: \(\frac{x^{16}}{16} + \frac{x^{13}}{13} + \frac{x^{10}}{10} + \frac{x^7}{7} + \frac{x^4}{4} + x\).
Breaking down each term is beneficial since more complex terms might require substitution or parts, but not in this case.
Constant of Integration
The constant of integration, represented by \(C\), is a vital part of indefinite integrals. It accounts for the fact that there are infinitely many antiderivatives for a function. Adding \(C\) establishes the general form of the antiderivative.

  • When you find the antiderivative of a function, it includes all possible vertical translations of that function.
  • The constant \(C\) represents these translations as it adds or subtracts a real number to every solution.
  • Skipping the constant means you could miss a solution that solves a particular condition or boundary in initial problems.
In solving any indefinite integral problem, remember to include \(C\) to represent this essential component.

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