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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 8.3.64
Capitolo 8, Problema 8.3.64

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ sin(θ) sin(2θ) sin(3θ) dθ

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Recognize that the integral involves a product of sine functions: \(\sin(\theta)\), \(\sin(2\theta)\), and \(\sin(3\theta)\). To simplify this, use trigonometric product-to-sum identities to rewrite the product of sines as sums of cosines or sines, which are easier to integrate.
Start by pairing two of the sine terms, for example, use the identity for the product of two sines: \(\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)]\). Apply this to \(\sin(\theta) \sin(2\theta)\).
After rewriting \(\sin(\theta) \sin(2\theta)\) as a sum of cosines, multiply the result by \(\sin(3\theta)\), and then apply product-to-sum identities again to simplify the product of sine and cosine terms.
Once the integrand is expressed as a sum of sine and cosine functions with single arguments, write the integral as the sum of integrals of these simpler trigonometric functions.
Integrate each term separately using the basic integral formulas: \(\int \sin(k\theta) d\theta = -\frac{1}{k} \cos(k\theta) + C\) and \(\int \cos(k\theta) d\theta = \frac{1}{k} \sin(k\theta) + C\). Combine all results to express the integral in its simplified form.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They allow simplification of complex expressions, such as products of sines and cosines, into sums or differences, making integration more manageable. Common identities include product-to-sum and double-angle formulas.
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Verifying Trig Equations as Identities

Product-to-Sum Formulas

Product-to-sum formulas convert products of sine and cosine functions into sums or differences of trigonometric functions. For example, sin A sin B = ½[cos(A−B) − cos(A+B)]. These formulas simplify integrals involving products of trigonometric functions by breaking them into sums that are easier to integrate.
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The Product Rule

Integration of Trigonometric Functions

Integrating trigonometric functions involves applying known antiderivatives and using identities to simplify the integrand. After rewriting products as sums, each term can be integrated using standard formulas, such as ∫sin(kθ)dθ = −(1/k)cos(kθ) + C. Mastery of these techniques is essential for solving integrals involving multiple trig functions.
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Introduction to Trigonometric Functions
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