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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 5.3.55
Chapter 5, Problem 5.3.55

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus


∫π/₄^³π/⁴ (cot² 𝓍 + 1) d𝓍

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Step 1: Recognize that the integral ∫π/₄^³π/⁴ (cot² 𝓍 + 1) d𝓍 can be simplified using trigonometric identities. Recall that cot²(𝓍) + 1 = csc²(𝓍). This simplifies the integral to ∫π/₄^³π/⁴ csc²(𝓍) d𝓍.
Step 2: Identify the antiderivative of csc²(𝓍). The antiderivative of csc²(𝓍) is -cot(𝓍). Using this, rewrite the integral as [-cot(𝓍)] evaluated from π/4 to 3π/4.
Step 3: Apply the Fundamental Theorem of Calculus. Substitute the upper limit (𝓍 = 3π/4) and the lower limit (𝓍 = π/4) into the antiderivative -cot(𝓍). This gives -cot(3π/4) - (-cot(π/4)).
Step 4: Simplify the expression. Use the unit circle or trigonometric properties to evaluate cot(3π/4) and cot(π/4). Recall that cot(π/4) = 1 and cot(3π/4) = -1.
Step 5: Combine the results from the previous step to find the value of the definite integral. This involves subtracting the evaluated terms from Step 4.

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Key Concepts

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

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 [a, b], then the integral of its derivative over that interval equals the difference in the values of the function at the endpoints. This theorem allows us to evaluate definite integrals by finding an antiderivative of the integrand.
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Fundamental Theorem of Calculus Part 1

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the limits of integration, which specify the interval, and provides a numerical value that reflects the accumulation of quantities, such as area, over that interval.
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Definition of the Definite Integral

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. In the context of the given integral, recognizing that cot² x + 1 equals csc² x can simplify the integration process, making it easier to evaluate the integral by transforming it into a more manageable form.
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Related Practice
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.                                                                                                                                      

                                                                                                                                                                                       

 ∫₀⁴ √(16― 𝓍² ) d𝓍

Textbook Question

Derivatives of integrals Simplify the following expressions.


d/dz ∫¹⁰ₛᵢₙ ₂ dt /(t⁴ + 1)

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Textbook Question

The composite function ƒ(g(𝓍)) consists of an inner function g and an outer function ƒ. If an integrand includes ƒ(g(𝓍)), which function is often a likely choice for a new variable u?

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Textbook Question

Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval.


ƒ(𝓍) = 8 ― 2𝓍 on [0, 4]

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Textbook Question

Variations on the substitution method Evaluate the following integrals.                                                                                                        

                                                                                                                                                                    

 ∫ 𝓍/(∛𝓍 + 4) d𝓍

Textbook Question

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    

                                                                                                                                                                    

  ∫ 𝓍 sin⁴ 𝓍² cos 𝓍² d𝓍 (Hint: Begin with u = 𝓍², and then use v = sin u .)