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.
18. ∫ dx / (225 − 16x²)

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Step 1: Recognize the integral's form. The given integral is ∫ dx / (225 − 16x²). This resembles the standard form ∫ dx / (a² − x²), which can be found in a table of integrals.

Step 2: Rewrite the denominator to match the standard form. Factor out the constant 16 from the denominator: 225 − 16x² = 15² − (4x)². This gives us ∫ dx / (15² − (4x)²).

Step 3: Perform a substitution to simplify the integral. Let u = 4x, so that du = 4 dx or dx = du / 4. Substitute these into the integral: ∫ dx / (15² − (4x)²) becomes (1/4) ∫ du / (15² − u²).

Step 4: Use the table of integrals for the standard form ∫ dx / (a² − x²). The result for this form is (1 / (2a)) ln |(a + x) / (a − x)| + C, where a is the constant and x is the variable. Apply this formula to the integral with a = 15 and u as the variable.

Step 5: Substitute back u = 4x into the result to express the solution in terms of x. Simplify the expression and include the constant of integration, C.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Indefinite Integrals

Indefinite integrals represent the family of antiderivatives of a function, expressed without specific limits. They are denoted by the integral sign followed by the function and 'dx', indicating integration with respect to x. The result includes a constant of integration, C, since antiderivatives are determined up to an additive constant.

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 identify standard forms in integral tables. For example, the expression 225 - 16x² can be rewritten to facilitate integration.

Integral Tables

Integral tables are collections of standard integrals that provide quick references for evaluating common integrals. They list integrals alongside their corresponding results, allowing for efficient computation without deriving each integral from first principles. Familiarity with these tables is essential for solving integrals that match standard forms, especially after transformations like completing the square.

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