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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.5.54
Chapter 5, Problem 5.5.54

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         
                                                                                                                                                                              
 ∫π/₁₆^π/⁸ 8 csc² 4𝓍 d𝓍

Verified step by step guidance
1
Step 1: Recognize that the integral involves the function csc²(4𝓍), which has a standard antiderivative. The antiderivative of csc²(u) is -cot(u).
Step 2: Perform a substitution to simplify the integral. Let u = 4𝓍, which implies that du = 4 d𝓍. Rewrite the integral in terms of u.
Step 3: Adjust the limits of integration according to the substitution. When 𝓍 = π/₁₆, u = 4(π/₁₆) = π/₄. When 𝓍 = π/₈, u = 4(π/₈) = π/₂.
Step 4: Rewrite the integral using the substitution. The integral becomes ∫π/₄^π/₂ 2 csc²(u) du, where the factor of 2 comes from dividing by 4 in the substitution.
Step 5: Evaluate the integral using the antiderivative of csc²(u). Substitute the limits of integration into -2 cot(u) and simplify.

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Key Concepts

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

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫[a, b] f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values over the interval [a, b].
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Change of Variables

Change of variables, or substitution, is a technique used in integration to simplify the integral by transforming it into a more manageable form. This involves substituting a new variable for an existing one, which can make the integral easier to evaluate. The process requires adjusting the limits of integration and the differential accordingly to maintain the integrity of the integral.
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Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). In calculus, it often appears in integrals involving trigonometric functions. Understanding its properties and behavior is essential for evaluating integrals that include csc²(x), which is commonly encountered in integration problems.
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Related Practice
Textbook Question

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The right Riemann sum for ƒ(𝓍)) = x + 1 on [0, 4] with n = 50.

Textbook Question

Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals

∫(e³ˣ ⁺¹ d𝓍

Textbook Question

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.                                                                                                                                      

                                                                                                                                                                                       

 ∫₋₁² ( ―|𝓍| ) d𝓍

Textbook Question

Derivatives of integrals Simplify the following expressions.


d/dy ∫¹⁰ᵧ³ √(𝓍⁶ + 1) d𝓍

Textbook Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 1/(𝓍² + 1) on [―1, 1]

Textbook Question

Areas of regions Find the area of the following regions.                                                                                                                   

                                                                                                                                                                 The region bounded by the graph of ƒ(𝓍) = (𝓍―4)⁴ and the 𝓍-axis between and 𝓍 = 2 and 𝓍= 6

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