Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(x³ + 1) / (x³ − x)] dx

검증된 단계별 안내

Start by examining the integrand \( \frac{x^3 + 1}{x^3 - x} \) to see if it can be simplified or if a substitution is appropriate.

Factor the denominator \( x^3 - x \) as \( x(x^2 - 1) \), and further factor \( x^2 - 1 \) as \( (x - 1)(x + 1) \). So the denominator becomes \( x(x - 1)(x + 1) \).

Consider performing polynomial division if the degree of the numerator is greater than or equal to the degree of the denominator, or try to express the integrand as a sum of partial fractions.

Set up the partial fraction decomposition for \( \frac{x^3 + 1}{x(x - 1)(x + 1)} \) as \( \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 1} \), and solve for constants \( A, B, C \) by multiplying both sides by the denominator and equating coefficients.

Once the partial fractions are found, integrate each term separately using the basic integral formulas \( \int \frac{1}{x} dx = \ln|x| + C \) and similar for the other terms.

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주요 개념

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Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand whose derivative also appears in the integral, allowing the integral to be rewritten in terms of a new variable. This technique is especially useful when the integral contains composite functions.

Partial Fraction Decomposition

Partial fraction decomposition breaks down a complex rational function into simpler fractions that are easier to integrate. This method requires factoring the denominator and expressing the integrand as a sum of simpler rational expressions. It is essential when dealing with integrals of rational functions where direct integration is difficult.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors, which simplifies the integrand and aids in methods like partial fraction decomposition. Recognizing common factors or special products helps to rewrite the integral in a more manageable form, making the integration process more straightforward.

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.∫ [(x³ + 1) / (x³ − x)] dx