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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 5.5.16d
Chapter 5, Problem 5.5.16d

Use Table 5.6 to evaluate the following definite integrals.                                                                                                                    
 (d) ∫₀^π/¹⁶ sec ² 4𝓍 d𝓍

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1
Step 1: Recognize the integral ∫₀^(π/16) sec²(4𝓍) d𝓍 as a standard form integral. The integral of sec²(u) with respect to u is tan(u).
Step 2: Identify the substitution needed for 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 based on the substitution. When 𝓍 = 0, u = 4(0) = 0. When 𝓍 = π/16, u = 4(π/16) = π/4.
Step 4: Rewrite the integral using the substitution: ∫₀^(π/16) sec²(4𝓍) d𝓍 becomes (1/4) ∫₀^(π/4) sec²(u) du, where the factor 1/4 comes from the substitution du = 4 d𝓍.
Step 5: Evaluate the integral ∫₀^(π/4) sec²(u) du using the antiderivative of sec²(u), which is tan(u). Substitute the limits of integration into tan(u) and multiply by the factor 1/4 to complete the solution.

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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]. Understanding how to evaluate definite integrals is crucial for solving problems in calculus.
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Definition of the Definite Integral

Integration Techniques

Integration techniques are methods used to find the integral of a function, especially when it cannot be integrated using basic formulas. Common techniques include substitution, integration by parts, and using trigonometric identities. In this context, recognizing the appropriate technique to apply is essential for evaluating the integral effectively, particularly when dealing with functions like sec²(4x).
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Trigonometric Functions

Trigonometric functions, such as secant (sec), are fundamental in calculus and relate angles to ratios of sides in right triangles. The secant function is defined as the reciprocal of the cosine function, sec(x) = 1/cos(x). Understanding the properties and derivatives of trigonometric functions is vital for evaluating integrals involving these functions, as they often appear in various calculus problems.
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Related Practice
Textbook Question

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(d) Assuming the velocity remains 10 m/s, for t ≥ 5, find the function that gives the displacement between t = 0 and any time t ≥ 5.

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Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.


∫₁⁷ 1/𝓍 d𝓍 ; n = 6

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Textbook Question

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.


{Use of Tech} ƒ(𝓍) = √x on [1,3] ; n = 4


(d) Calculate the midpoint Riemann sum.

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Textbook Question

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.

ƒ(𝓍) = x² ─ 1 on [2,4]; n = 4

(d) Calculate the left and right Riemann sums. 

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Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(c) Calculate the left and right Riemann sums for the given value of n.


∫₀² (𝓍²―2) d𝓍 ; n = 4

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Textbook Question

Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.

(a) ∫₀³ 5ƒ(𝓍) d𝓍

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