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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 8 - Techniques of Integration문제 8.1.12
8장, 문제 8.1.12

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫₋₁³ (4x² - 7) / (2x + 3) dx

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Start by examining the integrand \( \frac{4x^{2} - 7}{2x + 3} \). Since the degree of the numerator (2) is greater than the degree of the denominator (1), perform polynomial long division to simplify the integrand.
Divide \(4x^{2} - 7\) by \(2x + 3\) to express the integrand as a polynomial plus a proper rational function. This will give you an expression of the form \(Q(x) + \frac{R(x)}{2x + 3}\), where \(Q(x)\) is a polynomial and \(R(x)\) is a polynomial of degree less than 1.
Rewrite the integral as the sum of two integrals: \(\int_{-1}^{3} Q(x) \, dx + \int_{-1}^{3} \frac{R(x)}{2x + 3} \, dx\). This breaks the problem into simpler parts.
For the integral involving \(\frac{R(x)}{2x + 3}\), use substitution. Let \(u = 2x + 3\), then compute \(du = 2 \, dx\) or \(dx = \frac{du}{2}\). Change the limits of integration accordingly from \(x\) to \(u\).
Integrate both parts separately: the polynomial integral is straightforward, and the integral involving \(\frac{1}{u}\) will result in a natural logarithm function. After integrating, substitute back to the original variable \(x\) and evaluate the definite integral using the given limits.

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주요 개념

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

Polynomial Division

When integrating a rational function where the numerator's degree is equal to or higher than the denominator's, polynomial division simplifies the integrand into a polynomial plus a proper fraction. This makes the integral easier to evaluate by breaking it into simpler parts.
추천 영상:
07:00
Taylor Polynomials

Substitution Method

Substitution involves changing variables to simplify an integral, especially when the integrand contains a composite function. By letting u equal a part of the integrand (like the denominator), the integral can be transformed into a more straightforward form.
추천 영상:
07:33
Euler's Method

Definite Integral Evaluation

After finding the antiderivative, evaluating a definite integral requires applying the Fundamental Theorem of Calculus by substituting the upper and lower limits into the antiderivative and subtracting. This yields the exact area under the curve between the given bounds.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral
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