23-64. Integration Evaluate the following integrals.
47. ∫ (x³ - 10x² + 27x)/(x² - 10x + 25) dx

검증된 단계별 안내

First, observe the integral \( \int \frac{x^{3} - 10x^{2} + 27x}{x^{2} - 10x + 25} \, dx \). Notice that the denominator \( x^{2} - 10x + 25 \) can be factored or recognized as a perfect square. Specifically, \( x^{2} - 10x + 25 = (x - 5)^{2} \).

Next, perform polynomial division because the degree of the numerator (3) is higher than the degree of the denominator (2). Divide \( x^{3} - 10x^{2} + 27x \) by \( (x - 5)^{2} \) to rewrite the integrand as a polynomial plus a proper rational function.

After the division, express the integrand as \( Q(x) + \frac{R(x)}{(x - 5)^{2}} \), where \( Q(x) \) is the quotient polynomial and \( R(x) \) is the remainder polynomial with degree less than 2.

Then, split the integral into two parts: \( \int Q(x) \, dx + \int \frac{R(x)}{(x - 5)^{2}} \, dx \). The first integral is straightforward since \( Q(x) \) is a polynomial.

For the second integral, use substitution \( u = x - 5 \) to simplify the denominator and then integrate the resulting rational function by breaking it into simpler terms if necessary.

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.

Integration of Rational Functions

Integrating rational functions often involves breaking them into simpler parts, such as polynomials and proper fractions. After simplification, standard integration techniques like substitution or partial fractions can be applied to evaluate the integral.

Recognizing Perfect Square Trinomials

Identifying perfect square trinomials in the denominator, such as (x - 5)², helps simplify the integral. This recognition allows substitution methods or rewriting the integrand in a form that is easier to integrate.

23-64. Integration Evaluate the following integrals.47. ∫ (x³ - 10x² + 27x)/(x² - 10x + 25) dx