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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 5.5.92
Chapter 5, Problem 5.5.92

Integrals with sinΒ² 𝓍 and cosΒ² 𝓍 Evaluate the following integrals.                                                                                                             
                                                                                                                                                                    
 βˆ« 𝓍 cos²𝓍² d𝓍

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Step 1: Recognize that the integral involves a trigonometric function squared, specifically cosΒ²(𝓍²). To simplify, use the trigonometric identity cosΒ²(u) = (1 + cos(2u)) / 2, where u = 𝓍² in this case.
Step 2: Substitute the identity into the integral. The integral becomes ∫ 𝓍 * (1 + cos(2𝓍²)) / 2 d𝓍. Split the integral into two parts: ∫ 𝓍/2 d𝓍 + ∫ 𝓍 * cos(2𝓍²)/2 d𝓍.
Step 3: For the first term, ∫ 𝓍/2 d𝓍, integrate directly using the power rule for integration: ∫ 𝓍^n dx = (𝓍^(n+1)) / (n+1). This gives (𝓍² / 4).
Step 4: For the second term, ∫ 𝓍 * cos(2𝓍²)/2 d𝓍, use substitution. Let u = 2𝓍², so du = 4𝓍 d𝓍. Rewrite the integral in terms of u: (1/8) ∫ cos(u) du. The integral of cos(u) is sin(u), so this term becomes (1/8) sin(2𝓍²).
Step 5: Combine the results from both terms. The final expression for the integral is (𝓍² / 4) + (1/8) sin(2𝓍²) + C, where C is the constant of integration.

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Key identities include the Pythagorean identities, such as sinΒ²x + cosΒ²x = 1, and double angle formulas. These identities are essential for simplifying integrals involving sinΒ²x and cosΒ²x, allowing for easier evaluation.
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Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric substitution. For integrals involving trigonometric functions like cosΒ²x, recognizing patterns and applying appropriate techniques is crucial for finding the antiderivative.
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Definite and Indefinite Integrals

Definite integrals calculate the area under a curve between two specified limits, while indefinite integrals represent a family of functions without limits. Understanding the difference is important when evaluating integrals, as it affects the final result. In this context, knowing whether to apply limits or find a general antiderivative is key to solving the problem.
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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 Ζ’(𝓍)) = x + 1 on [0, 4] with n = 50.

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Textbook Question

Derivatives of integrals Simplify the following expressions.


d/dt βˆ«β‚€α΅— d𝓍/(1 + 𝓍²) + βˆ«β‚ΒΉ/α΅— dx/(1 + 𝓍²)

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Textbook Question

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function Ζ’ on [a,b]. Identify Ζ’ and express the limit as a definite integral.                                

          n                                                                                                                                                                              

    lim   βˆ‘ (𝓍ₖ*Β² + 1) βˆ†π“β‚– on [0,2]                                                                                                                                                                            

  βˆ† β†’ 0   k=1                                                                                                                                                                                                                      

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Textbook Question

Area Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question.


The region bounded by y = 6 cos 𝓍 and the 𝓍-axis between 𝓍 = ―π/2 and 𝓍 = Ο€

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Textbook Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

                                                                                                                                                                              

 βˆ«β‚‹β‚ΒΉ (𝓍―1) (𝓍²―2𝓍)⁷ d𝓍

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Textbook Question

Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.


βˆ«α΅ƒβ‚‹β‚ Ζ’(g(𝓍)) d𝓍

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