Domanda del libro di testo
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ sin(2x) cos(4x) dx
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 8.6.22
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ sin(2x) cos(3x) dx
Guida verificata passo dopo passo1
Recognize that the integral involves the product of sine and cosine functions with different arguments: \(\int \sin(2x) \cos(3x) \, dx\).
Use the product-to-sum trigonometric identity to rewrite the product of sine and cosine as a sum of sines: \(\sin(A) \cos(B) = \frac{1}{2} [\sin(A+B) + \sin(A-B)]\).
Apply the identity with \(A = 2x\) and \(B = 3x\) to get: \(\sin(2x) \cos(3x) = \frac{1}{2} [\sin(5x) + \sin(-x)]\).
Simplify \(\sin(-x)\) using the odd property of sine: \(\sin(-x) = -\sin(x)\), so the expression becomes \(\frac{1}{2} [\sin(5x) - \sin(x)]\).
Rewrite the integral as \(\int \sin(2x) \cos(3x) \, dx = \frac{1}{2} \int [\sin(5x) - \sin(x)] \, dx\), then integrate each sine term separately using the integral formula \(\int \sin(kx) \, dx = -\frac{1}{k} \cos(kx) + C\).
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Product-to-Sum Trigonometric Identities
These identities transform products of sine and cosine functions into sums or differences of trigonometric functions, simplifying integration. For example, sin(A)cos(B) = ½[sin(A+B) + sin(A−B)]. Using these identities helps convert the integral into a form easier to integrate.
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Verifying Trig Equations as Identities
Basic Integration of Trigonometric Functions
Integrating sine and cosine functions involves applying standard integral formulas, such as ∫sin(kx) dx = −(1/k)cos(kx) + C and ∫cos(kx) dx = (1/k)sin(kx) + C. Recognizing these forms allows direct evaluation after simplification.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Use of Integral Tables
Integral tables provide formulas for common integrals, including trigonometric integrals, which can save time and reduce errors. Referring to these tables helps identify the correct integral form and constants, especially for more complex expressions.
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Percorso guidato
08:01Integration Using Partial Fractions
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