7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
11. ∫ 3u / (2u + 7) du

Verified step by step guidance

1. Step 1: Begin by analyzing the integral ∫ (3u / (2u + 7)) du. Notice that the denominator is linear (2u + 7). This suggests that a substitution might simplify the integral.
2. Step 2: Perform a substitution. Let v = 2u + 7, which implies dv/du = 2 or du = dv/2. Rewrite the integral in terms of v.
3. Step 3: Substitute into the integral. The numerator 3u can be expressed in terms of v using the substitution u = (v - 7)/2. Replace u and du in the integral.
4. Step 4: Simplify the integral after substitution. You should now have an expression involving v that can be matched to a standard form in a table of integrals.
5. Step 5: Use the table of integrals to find the antiderivative of the simplified expression. After finding the antiderivative, substitute back v = 2u + 7 to return to the original variable u.

Key Concepts

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. Understanding how to evaluate indefinite integrals is crucial for solving problems in calculus, as they provide the antiderivative of a function.

Integration Techniques

Various techniques exist for evaluating integrals, including substitution, integration by parts, and using tables of integrals. In this case, recognizing when to apply substitution or complete the square is essential for transforming the integrand into a form that can be easily looked up in a table of integrals.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in a specific form, which can simplify integration. This technique involves rearranging a quadratic expression into a perfect square trinomial plus a constant, making it easier to integrate functions that involve quadratics, especially when using tables of integrals.