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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.6.26
Chapter 8, Problem 8.6.26

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ cos(θ / 2) cos(7θ) dθ

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1
Recognize that the integral involves the product of two cosine functions: \(\cos\left(\frac{\theta}{2}\right)\) and \(\cos(7\theta)\). To simplify this, recall the product-to-sum identity for cosine: \(\cos A \cos B = \frac{1}{2} [\cos(A - B) + \cos(A + B)]\).
Apply the product-to-sum formula by letting \(A = \frac{\theta}{2}\) and \(B = 7\theta\). Substitute these into the identity to rewrite the integrand as \(\frac{1}{2} \left[ \cos\left(\frac{\theta}{2} - 7\theta\right) + \cos\left(\frac{\theta}{2} + 7\theta\right) \right]\).
Simplify the arguments inside the cosine functions: \(\frac{\theta}{2} - 7\theta = \frac{\theta}{2} - \frac{14\theta}{2} = -\frac{13\theta}{2}\) and \(\frac{\theta}{2} + 7\theta = \frac{\theta}{2} + \frac{14\theta}{2} = \frac{15\theta}{2}\).
Rewrite the integral as \(\int \cos\left(\frac{\theta}{2}\right) \cos(7\theta) \, d\theta = \frac{1}{2} \int \left[ \cos\left(-\frac{13\theta}{2}\right) + \cos\left(\frac{15\theta}{2}\right) \right] d\theta\). Remember that \(\cos(-x) = \cos x\), so you can simplify the first cosine term.
Split the integral into two separate integrals: \(\frac{1}{2} \left( \int \cos\left(\frac{13\theta}{2}\right) d\theta + \int \cos\left(\frac{15\theta}{2}\right) d\theta \right)\). Use the table of integrals to find the antiderivatives of each cosine term, which generally have the form \(\int \cos(k\theta) d\theta = \frac{1}{k} \sin(k\theta) + C\).

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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, such as product-to-sum formulas, allow the transformation of products of trigonometric functions into sums or differences. This simplification is essential for integrating products like cos(θ/2) cos(7θ), making the integral easier to evaluate using standard formulas.
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Use of Integral Tables

Integral tables provide formulas for common integrals, including those involving trigonometric functions. Knowing how to locate and apply these formulas helps efficiently evaluate integrals without performing lengthy derivations, especially when the integrand matches a standard form.
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Integration of Trigonometric Functions

Integrating trigonometric functions involves applying known antiderivatives and sometimes using substitution or identities. Understanding the basic integrals of sine and cosine functions and how to handle their arguments is crucial for solving integrals like ∫ cos(θ/2) cos(7θ) dθ.
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Related Practice
Textbook Question

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ cos^(-1)(√x) / √x dx

Textbook Question

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (1 - x²)^(1/2) / x⁴ dx

Textbook Question

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ 1 / (x⁴ + x) dx

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