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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.5.92
Capitolo 5, Problema 5.5.92

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             
                                                                                                                                                                    
 ∫ 𝓍 cos²𝓍² d𝓍

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Step 1: Recognize that the integral involves a trigonometric function squared, specifically cos²(𝓍²). To simplify, use the trigonometric identity cos²(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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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 sin²x + cos²x = 1, and double angle formulas. These identities are essential for simplifying integrals involving sin²x and cos²x, 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 cos²x, recognizing patterns and applying appropriate techniques is crucial for finding the antiderivative.
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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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