7–64. Integration review Evaluate the following integrals.
62. ∫ (-x⁵ - x⁴ - 2x³ + 4x + 3) / (x² + x + 1) dx

Verified step by step guidance

1. Step 1: Begin by analyzing the integrand. The numerator is a polynomial of degree 5, and the denominator is a polynomial of degree 2. Since the degree of the numerator is higher than the degree of the denominator, perform polynomial long division to simplify the integrand.
2. Step 2: Divide the numerator (-x⁵ - x⁴ - 2x³ + 4x + 3) by the denominator (x² + x + 1). The result will be a quotient (a polynomial) and a remainder. Write the integrand as the sum of the quotient and the remainder divided by the denominator.
3. Step 3: After performing the division, the integral will be split into two parts: the integral of the quotient and the integral of the remainder divided by the denominator. Focus on solving each part separately.
4. Step 4: For the integral of the quotient, integrate term by term using basic power rule integration: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1.
5. Step 5: For the integral of the remainder divided by the denominator, consider whether partial fraction decomposition or substitution is necessary. Simplify the remainder term and proceed with integration techniques such as substitution or recognizing standard integral forms.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It can be thought of as the reverse process of differentiation. In this context, we are tasked with evaluating a specific integral, which requires applying techniques such as polynomial long division or substitution.

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to numerical long division. This technique is particularly useful when the degree of the numerator is greater than the degree of the denominator, as it simplifies the integrand into a more manageable form. By performing this division, we can separate the integral into simpler parts that are easier to evaluate.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given integral, the integrand is a rational function, which often requires specific techniques for integration, such as partial fraction decomposition or polynomial long division. Understanding the properties of rational functions is crucial for effectively evaluating integrals involving them.