Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(2x³ + x² − 21x + 24) / (x² + 2x − 8)] dx

검증된 단계별 안내

First, observe the integral \( \int \frac{2x^3 + x^2 - 21x + 24}{x^2 + 2x - 8} \, dx \). Since the degree of the numerator (3) is higher than the degree of the denominator (2), start by performing polynomial long division to simplify the integrand.

Divide \( 2x^3 + x^2 - 21x + 24 \) by \( x^2 + 2x - 8 \) to express the integrand as a polynomial plus a proper rational function (where the numerator degree is less than the denominator degree).

After the division, rewrite the integral as \( \int (\text{quotient} + \frac{\text{remainder}}{x^2 + 2x - 8}) \, dx \). Then, split the integral into two parts: one involving the polynomial quotient and the other involving the proper rational function.

Next, factor the denominator \( x^2 + 2x - 8 \) into \( (x + 4)(x - 2) \) to prepare for partial fraction decomposition of the proper rational function.

Set up the partial fraction decomposition for the proper rational function as \( \frac{A}{x + 4} + \frac{B}{x - 2} \), solve for constants \( A \) and \( B \), then integrate each term separately.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Polynomial Long Division

When the degree of the numerator is equal to or greater than the denominator in a rational function, polynomial long division is used to simplify the integrand. This process rewrites the integrand as a polynomial plus a proper fraction, making the integral easier to evaluate.

Substitution Method

Substitution involves changing variables to simplify an integral, often by letting a part of the integrand equal a new variable. This technique is useful when the integral contains a composite function or when the derivative of a function appears elsewhere in the integrand.

Partial Fraction Decomposition

Partial fraction decomposition breaks a rational function into simpler fractions that are easier to integrate. After ensuring the fraction is proper, the denominator is factored, and the integrand is expressed as a sum of fractions with unknown coefficients to be determined.

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.∫ [(2x³ + x² − 21x + 24) / (x² + 2x − 8)] dx