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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.3.6
Capitolo 8, Problema 8.3.6

Evaluate the integrals in Exercises 1–22.
∫ cos³(4x) dx

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1
Recognize that the integral involves a power of cosine: \(\int \cos^{3}(4x) \, dx\). To simplify, use the identity for odd powers of cosine: express \(\cos^{3}(\theta)\) as \(\cos(\theta) \cdot \cos^{2}(\theta)\).
Rewrite \(\cos^{2}(\theta)\) using the Pythagorean identity: \(\cos^{2}(\theta) = 1 - \sin^{2}(\theta)\). So, \(\cos^{3}(4x) = \cos(4x) \cdot (1 - \sin^{2}(4x))\).
Substitute this back into the integral: \(\int \cos(4x) (1 - \sin^{2}(4x)) \, dx = \int \cos(4x) \, dx - \int \cos(4x) \sin^{2}(4x) \, dx\).
Use substitution for the second integral: let \(u = \sin(4x)\), then \(du = 4 \cos(4x) \, dx\), or equivalently, \(\cos(4x) \, dx = \frac{du}{4}\). Rewrite the integral in terms of \(u\).
Integrate the resulting expression in \(u\), then substitute back \(u = \sin(4x)\) to express the answer in terms of \(x\). Don't forget to add the constant of integration \(C\).

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. For integrals like ∫ cos³(4x) dx, identities such as the power-reduction or product-to-sum formulas help rewrite powers of cosine into expressions easier to integrate.
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Verifying Trig Equations as Identities

Integration of Powers of Trigonometric Functions

Integrating powers of sine or cosine often requires reducing the power using identities or substitution. For odd powers, one factor is separated, and the remaining even power is converted using identities, simplifying the integral into basic trigonometric integrals.
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Introduction to Trigonometric Functions

Substitution Method

The substitution method involves changing variables to simplify an integral. For example, when integrating functions like cos³(4x), substituting u = 4x transforms the integral into a simpler form, making it easier to apply standard integration techniques.
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Euler's Method
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