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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 8.3.62
Chapter 8, Problem 8.3.62

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ sin³(θ) cos(2θ) dθ

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Recognize that the integral involves powers of sine and cosine with different arguments, so start by expressing \( \sin^3(\theta) \) in a form that is easier to integrate. Use the identity \( \sin^3(\theta) = \sin(\theta) \cdot \sin^2(\theta) \) and then rewrite \( \sin^2(\theta) \) using the Pythagorean identity \( \sin^2(\theta) = 1 - \cos^2(\theta) \).
Rewrite the integral as \( \int \sin(\theta) (1 - \cos^2(\theta)) \cos(2\theta) \, d\theta \). This separates the powers and prepares the integral for substitution or further simplification.
Use the double-angle identity for cosine: \( \cos(2\theta) = 2\cos^2(\theta) - 1 \). Substitute this into the integral to express everything in terms of \( \cos(\theta) \) and \( \sin(\theta) \).
Consider the substitution \( u = \cos(\theta) \), which implies \( du = -\sin(\theta) d\theta \). Rewrite the integral in terms of \( u \) and \( du \), replacing \( \sin(\theta) d\theta \) with \( -du \).
After substitution, the integral becomes a polynomial in \( u \). Expand and simplify the polynomial expression, then integrate term-by-term with respect to \( u \).

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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 are equations involving trigonometric functions that hold true for all values within their domains. They simplify expressions and integrals by rewriting powers or products of sine and cosine into more manageable forms, such as using double-angle or power-reduction formulas.
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Integration of Trigonometric Functions

Integrating trigonometric functions often requires rewriting the integrand using identities to express it in terms of basic sine and cosine functions. This process makes it easier to apply standard integral formulas and find antiderivatives.
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Power-Reducing and Double-Angle Formulas

Power-reducing formulas convert powers of sine or cosine into expressions involving first powers of cosine or sine with multiple angles. Double-angle formulas express functions like cos(2θ) in terms of cos²θ and sin²θ, aiding in simplifying integrals involving products of trigonometric powers.
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Related Practice
Textbook Question

Evaluate the integrals in Exercises 33–52.

∫ sec³(x) tan³(x) dx

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.

∫ (x^2 + 6x) / (x^2 + 3)^2 dx

Textbook Question

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ (r² + r + 1) e^r dr

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In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.

∫₁^∞ (1 / (x² + 3x)) dx

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Textbook Question

Use the substitution u = tan x to evaluate the integral

∫ dx / (1 + sin² x).

Textbook Question

Expand the quotients in Exercises 1–8 by partial fractions.

(t⁴ + 9) / (t⁴ + 9t²)

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