7–64. Integration review Evaluate the following integrals.
32. ∫ from 0 to 2 of x / (x² + 4x + 8) dx

Verified step by step guidance

1. Step 1: Begin by analyzing the integrand, which is x / (x² + 4x + 8). Notice that the denominator is a quadratic expression. Factorization is not possible here, so consider completing the square for the quadratic expression x² + 4x + 8.
2. Step 2: Complete the square for x² + 4x + 8. Rewrite it as (x + 2)² + 4. This transformation helps simplify the denominator and makes it easier to work with.
3. Step 3: Perform a substitution to simplify the integral. Let u = x + 2, which implies du = dx. Also, adjust the limits of integration: when x = 0, u = 2; and when x = 2, u = 4.
4. Step 4: Rewrite the integral in terms of u. The integral becomes ∫ from u = 2 to u = 4 of (u - 2) / (u² + 4) du. Split the numerator into two terms: ∫ from u = 2 to u = 4 of u / (u² + 4) du - ∫ from u = 2 to u = 4 of 2 / (u² + 4) du.
5. Step 5: Solve each term separately. For the first term, use substitution v = u² + 4, dv = 2u du. For the second term, recognize it as a standard integral of the form ∫ 1 / (u² + a²) du, which evaluates to (1/a) arctan(u/a). Combine the results to express the solution.

Key Concepts

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫ from a to b of f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values over the interval [a, b].

Integration Techniques

Integration techniques are methods used to evaluate integrals that may not be solvable by basic antiderivatives. Common techniques include substitution, integration by parts, and partial fraction decomposition. For the integral in the question, recognizing the form of the denominator can guide the choice of technique to simplify the expression before integrating.

Polynomial Division and Completing the Square

Polynomial division and completing the square are algebraic techniques that can simplify integrals involving polynomials. Completing the square transforms a quadratic expression into a perfect square form, which can facilitate integration. In the given integral, rewriting the denominator x² + 4x + 8 in a completed square form can help in applying substitution or recognizing a standard integral form.