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.
31. ∫ √(x² - 8x) dx, x > 8

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Rewrite the expression inside the square root to make it easier to work with. Complete the square for the quadratic expression x² - 8x. Completing the square involves rewriting x² - 8x as (x - 4)² - 16.

Substitute the completed square form into the integral: ∫ √((x - 4)² - 16) dx.

Recognize that this integral resembles a standard form found in a table of integrals, specifically one involving √(u² - a²). To match the table form, let u = x - 4 and du = dx.

Rewrite the integral in terms of u: ∫ √(u² - 16) du. Now the integral is in a standard form that can be evaluated using a table of integrals.

Use the appropriate formula from the table of integrals for ∫ √(u² - a²) du, where a² = 16. Substitute back u = x - 4 into the result to express the solution in terms of x.

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Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This method is essential for simplifying integrals involving quadratic terms, as it allows for easier integration by rewriting the expression in a more manageable form. For example, the expression x² - 8x can be rewritten as (x - 4)² - 16, facilitating the integration process.

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative gives the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. Understanding how to evaluate indefinite integrals using techniques such as substitution or reference to integral tables is crucial for solving problems in calculus, including those that require preliminary transformations.

Integral Tables

Integral tables are collections of standard integrals that provide quick references for evaluating common integrals without performing the integration from first principles. They are particularly useful for complex functions or those that require specific techniques, such as trigonometric or exponential substitutions. Familiarity with these tables can significantly expedite the process of solving integrals, especially when combined with preliminary algebraic manipulations.

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.31. ∫ √(x² - 8x) dx, x > 8