Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

Problema 5.5.88

Capitolo 5, Problema 5.5.88

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.

∫ sin² 𝓍 d𝓍

Guida verificata passo dopo passo

Step 1: Recognize that the integral involves sin²(𝓍). To simplify this, use the trigonometric identity sin²(𝓍) = (1 - cos(2𝓍)) / 2.

Step 2: Rewrite the integral using the identity: ∫ sin²(𝓍) d𝓍 = ∫ [(1 - cos(2𝓍)) / 2] d𝓍.

Step 3: Split the integral into two parts: ∫ [(1/2) - (cos(2𝓍)/2)] d𝓍 = (1/2) ∫ 1 d𝓍 - (1/2) ∫ cos(2𝓍) d𝓍.

Step 4: Evaluate each part separately. For the first term, ∫ 1 d𝓍 = 𝓍. For the second term, use the formula for the integral of cos(k𝓍): ∫ cos(k𝓍) d𝓍 = (1/k) sin(k𝓍). Here, k = 2.

Step 5: Combine the results: (1/2)𝓍 - (1/4)sin(2𝓍) + 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 sin²(x) + cos²(x) = 1, which can simplify integrals involving sin²(x) and cos²(x). Understanding these identities is crucial for transforming integrals into more manageable forms.

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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 using trigonometric identities. For integrals involving sin²(x), the power-reduction formula can be particularly useful, allowing the integral to be expressed in terms of simpler functions that are easier to integrate.

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Integration by Parts for Definite Integrals

Definite and Indefinite Integrals

Definite integrals calculate the area under a curve between two specified limits, while indefinite integrals represent a family of functions and include a constant of integration. Understanding the difference is essential when evaluating integrals, as it affects the final result and the interpretation of the integral in a given context.

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