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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.5.28
Capitolo 8, Problema 8.5.28

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ 1 / (x⁴ + x) dx

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1
Start by factoring the denominator of the integrand. The denominator is \(x^{4} + x\), which can be factored by taking out the common factor \(x\): \[x^{4} + x = x(x^{3} + 1)\]
Next, factor the cubic term \(x^{3} + 1\) using the sum of cubes formula: \[a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\] Here, \(a = x\) and \(b = 1\), so \[x^{3} + 1 = (x + 1)(x^{2} - x + 1)\]
Rewrite the integrand using the full factorization: \[\frac{1}{x^{4} + x} = \frac{1}{x(x + 1)(x^{2} - x + 1)}\]
Set up the partial fraction decomposition for the integrand: \[\frac{1}{x(x + 1)(x^{2} - x + 1)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{Cx + D}{x^{2} - x + 1}\] where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.
Multiply both sides by the denominator \(x(x + 1)(x^{2} - x + 1)\) to clear the fractions, then equate coefficients of corresponding powers of \(x\) to form a system of equations. Solve this system to find the values of \(A\), \(B\), \(C\), and \(D\). Once found, integrate each term separately to evaluate the integral.

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Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a complex rational function as a sum of simpler fractions. This method is especially useful for integrating rational functions by breaking them into terms that are easier to integrate individually.
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Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. For partial fractions, factoring the denominator into linear and irreducible quadratic factors is essential to set up the correct form of the decomposition.
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Integration of Rational Functions

Integrating rational functions often requires rewriting the integrand into simpler parts via partial fractions. Each simpler fraction corresponds to a standard integral form, such as logarithmic or arctangent functions, facilitating the evaluation of the integral.
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Intro to Rational Functions
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