7–84. Evaluate the following integrals.
47. ∫ [(2x³ + x² - 2x - 4) / (x² - x - 2)] dx

Verified step by step guidance

1. Step 1: Factorize the denominator (x² - x - 2). Look for two numbers that multiply to -2 and add to -1. The factorization is (x - 2)(x + 1).
2. Step 2: Perform polynomial long division to simplify the fraction (2x³ + x² - 2x - 4) / (x² - x - 2). Divide the numerator by the denominator to express the integrand as a sum of simpler terms.
3. Step 3: After performing the division, the result will be a combination of a polynomial and a proper fraction. Decompose the proper fraction into partial fractions. For example, express A / (x - 2) + B / (x + 1) and solve for A and B.
4. Step 4: Write the integral as the sum of the integral of the polynomial term and the integral of the partial fractions. Use the rules of integration for polynomials and logarithmic functions.
5. Step 5: Integrate each term separately. For the polynomial term, use the power rule of integration. For the partial fractions, use the formula ∫(1 / (x - c)) dx = ln|x - c|. Combine all results to express the final solution.

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, evaluating the integral requires understanding how to manipulate and simplify the integrand before applying integration techniques.

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 techniques such as polynomial long division or partial fraction decomposition to simplify before integration. Understanding how to handle rational functions is crucial for evaluating the integral effectively.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to break down a complex rational function into simpler fractions that are easier to integrate. This technique is particularly useful when the degree of the numerator is less than the degree of the denominator. By expressing the integrand in this form, one can integrate each term separately, facilitating the evaluation of the integral.