Textbook Question
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
∫ sec 4w tan 4w dw
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7. Antiderivatives & Indefinite Integrals
Integrals of Trig Functions
2:11 minutesProblem 8.R.79
Textbook Question
76-81. Table of integrals Use a table of integrals to evaluate the following integrals.
79. ∫ sec⁵x dx
Verified step by step guidance1
Step 1: Recognize that the integral ∫ sec⁵x dx matches a form found in a table of integrals. Tables of integrals often provide solutions for powers of secant functions.
Step 2: Locate the formula in the table of integrals for ∫ sec⁵x dx. Typically, this formula involves breaking down the integral into simpler terms using reduction formulas or trigonometric identities.
Step 3: Apply the formula from the table of integrals. For example, the formula might express ∫ sec⁵x dx in terms of ∫ sec³x dx and ∫ secx dx, which are simpler integrals.
Step 4: Substitute the expressions for ∫ sec³x dx and ∫ secx dx into the formula, if required, and simplify the result. This step may involve algebraic manipulation or substitution.
Step 5: Add the constant of integration (C) to the final expression, as indefinite integrals always include an arbitrary constant.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration
Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It is the reverse process of differentiation and can be used to calculate quantities such as total distance, area, and volume. Understanding the basic rules and techniques of integration is essential for solving integral problems.
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Secant Function
The secant function, denoted as sec(x), is defined as the reciprocal of the cosine function: sec(x) = 1/cos(x). It is important in calculus, particularly in integration and differentiation, as it appears frequently in trigonometric integrals. Recognizing the properties and behavior of the secant function helps in evaluating integrals involving secant and its powers.
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Table of Integrals
A table of integrals is a reference tool that lists common integrals and their corresponding results. It simplifies the process of integration by providing ready-made solutions for frequently encountered functions, such as polynomials, trigonometric functions, and exponential functions. Familiarity with a table of integrals can significantly speed up the evaluation of complex integrals, like ∫ sec⁵x dx.
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