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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.R.45
Chapter 5, Problem 5.R.45

Evaluating integrals Evaluate the following integrals.


∫π/₆^π/³ (sec² t + csc² t) dt

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Step 1: Recognize that the integral involves the sum of two trigonometric functions: sec²(t) and csc²(t). Recall their respective antiderivatives: ∫sec²(t) dt = tan(t) and ∫csc²(t) dt = -cot(t).
Step 2: Break the integral into two separate integrals: ∫(sec²(t) + csc²(t)) dt = ∫sec²(t) dt + ∫csc²(t) dt.
Step 3: Compute the antiderivative of each term. For sec²(t), the antiderivative is tan(t). For csc²(t), the antiderivative is -cot(t). Combine these results to get tan(t) - cot(t).
Step 4: Apply the Fundamental Theorem of Calculus to evaluate the definite integral. Substitute the upper limit π/3 and the lower limit π/6 into the antiderivative expression tan(t) - cot(t).
Step 5: Simplify the result by calculating tan(π/3), tan(π/6), cot(π/3), and cot(π/6). Subtract the values obtained at the lower limit from those at the upper limit to find the final value of the integral.

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Key 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 accumulated area under a curve represented by a function. It is the reverse process of differentiation and is used to calculate quantities such as areas, volumes, and total accumulated change. The integral symbol (∫) denotes the operation, and definite integrals, like the one in the question, have specified limits of integration.
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Integration by Parts for Definite Integrals

Trigonometric Functions

Trigonometric functions, such as secant (sec) and cosecant (csc), are essential in calculus, particularly in integrals involving angles. The secant function is defined as the reciprocal of the cosine function, while the cosecant function is the reciprocal of the sine function. Understanding their properties and relationships is crucial for evaluating integrals that include these functions.
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Introduction to Trigonometric Functions

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration, stating that if a function is continuous on an interval, then the integral of its derivative over that interval equals the change in the function's values at the endpoints. This theorem allows us to evaluate definite integrals by finding an antiderivative of the integrand and applying the limits of integration, simplifying the process of calculating areas under curves.
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Fundamental Theorem of Calculus Part 1
Related Practice
Textbook Question

Evaluating integrals Evaluate the following integrals.


∫₋₂² (3𝓍⁴―2𝓍 + 1) d𝓍

Textbook Question

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.


(b) Use summation notation to express the right Riemann sum in terms of a positive integer n .

Textbook Question

Velocity to displacement An object travels on the 𝓍-axis with a velocity given by v(t) = 2t + 5, for 0 ≤ t ≤ 4.


(a) How far does the object travel, for 0 ≤ t ≤ 4 ?

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Textbook Question

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.


―∫₄¹ 2ƒ(𝓍) d𝓍

Textbook Question

Area functions and the Fundamental Theorem Consider the function

ƒ(t) = { t      if  ―2 ≤ t < 0

t²/2    if    0 ≤ t ≤ 2

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(b) Use the Fundamental Theorem to find an expression for F '(𝓍) for ―2 ≤ 𝓍 < 0.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(d) If ƒ is continuous on [a,b] and ∫ₐᵇ |ƒ(𝓍)| d𝓍 = 0 , then ƒ(𝓍) = 0 on [a,b] .

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