Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ ((2ˣ - 1) / 3ˣ) dx
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 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
Verified step by step guidance1
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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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 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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05:18The 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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6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ e^(-3t) sin(4t) dt
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞^∞ (x dx) / (x² + 4)^(3/2)
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Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ p⁴ e^(-p) dp
Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ (2 dx) / (x³ √(x² - 1)), where x > 1
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫₋₁³ (4x² - 7) / (2x + 3) dx
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