Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.
∫ (5x² + 18x + 20) / [(2x + 3)(x² + 4x + 8)] dx

Verified step by step guidance

1. Step 1: Analyze the structure of the integrand. The numerator is a polynomial (5x² + 18x + 20), and the denominator is a product of two factors: a linear term (2x + 3) and a quadratic term (x² + 4x + 8). This suggests that partial fraction decomposition may be a suitable technique.
2. Step 2: Set up the partial fraction decomposition. Express the integrand as a sum of fractions: A/(2x + 3) + (Bx + C)/(x² + 4x + 8), where A, B, and C are constants to be determined.
3. Step 3: Multiply through by the denominator [(2x + 3)(x² + 4x + 8)] to eliminate the fractions. This will result in an equation involving the numerator (5x² + 18x + 20) and the expanded terms from the partial fraction decomposition.
4. Step 4: Solve for the constants A, B, and C by equating coefficients of like powers of x on both sides of the equation. This involves algebraic manipulation to match terms.
5. Step 5: Once the partial fractions are determined, rewrite the integral as the sum of simpler integrals: ∫ A/(2x + 3) dx + ∫ (Bx + C)/(x² + 4x + 8) dx. Each term can then be integrated using appropriate techniques, such as substitution for the linear term and completing the square for the quadratic term.

Key Concepts

Integration Techniques

Integration techniques are methods used to evaluate integrals, which can include substitution, integration by parts, partial fraction decomposition, and trigonometric substitution. Each technique is suited for different types of integrands, and understanding when to apply each method is crucial for simplifying the integration process.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a rational function into simpler fractions that are easier to integrate. This method is particularly useful when the integrand is a fraction where the degree of the numerator is less than the degree of the denominator, allowing for straightforward integration of each term.

Simplifying the Integrand

Simplifying the integrand involves rewriting the expression in a form that makes integration easier. This can include factoring polynomials, canceling common terms, or using algebraic manipulation to express the integrand in a more manageable way, which can significantly aid in the selection of an appropriate integration technique.