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.
37. ∫ dx / √(x² + 10x), x >

검증된 단계별 안내

Step 1: Begin by completing the square for the expression inside the square root, x² + 10x. Rewrite it as (x² + 10x + 25 - 25), which simplifies to (x + 5)² - 25.

Step 2: Substitute u = x + 5 to simplify the integral. This substitution transforms the expression into ∫ dx / √(u² - 25), where u = x + 5 and du = dx.

Step 3: Recognize that the integral now matches a standard form found in a table of integrals: ∫ dx / √(u² - a²), which corresponds to the formula ln|u + √(u² - a²)| + C.

Step 4: Apply the formula from the table of integrals, substituting u = x + 5 and a² = 25 (so a = 5). The result is ln|u + √(u² - 25)| + C.

Step 5: Replace u with x + 5 to return to the original variable. The final expression becomes ln|x + 5 + √((x + 5)² - 25)| + C.

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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.

Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This method is particularly useful in integration, as it simplifies the integrand, making it easier to apply standard integral formulas. For example, the expression x² + 10x can be rewritten as (x + 5)² - 25, facilitating integration.

Integral Tables

Integral tables are collections of standard integrals that provide quick references for evaluating common integrals. They save time and effort by allowing students to look up the integral of a function rather than calculating it from scratch. Familiarity with these tables and knowing when to use them is essential for efficiently solving integral problems.

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.37. ∫ dx / √(x² + 10x), x >