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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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