Domanda del libro di testo
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (3t² + t + 4) / (t³ + t) dt from 1 to √3
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Ch. 8 - Techniques of Integration
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 8, Problema 8.3.54
Evaluate the integrals in Exercises 53–58.
∫ 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 identity for sine and cosine: \(\sin A \cos B = \frac{1}{2} [\sin(A+B) + \sin(A-B)]\).
Apply the identity with \(A = 2x\) and \(B = 3x\) to rewrite the integral as \(\int \sin(2x) \cos(3x) \, dx = \int \frac{1}{2} [\sin(2x + 3x) + \sin(2x - 3x)] \, dx\).
Simplify the arguments inside the sine functions: \(\sin(5x)\) and \(\sin(-x)\), so the integral becomes \(\frac{1}{2} \int [\sin(5x) + \sin(-x)] \, dx\).
Recall that \(\sin(-x) = -\sin x\), then split the integral into two simpler integrals: \(\frac{1}{2} \left( \int \sin(5x) \, dx - \int \sin x \, dx \right)\), and proceed to integrate each term separately.
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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) = 1/2 [sin(A+B) + sin(A−B)]. Using these identities helps convert the integral into a more manageable form.
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Verifying Trig Equations as Identities
Basic Integration of Sine and Cosine Functions
Integrating sine and cosine functions involves reversing differentiation: ∫sin(kx) dx = −(1/k)cos(kx) + C and ∫cos(kx) dx = (1/k)sin(kx) + C. Recognizing these formulas allows direct integration once the integrand is simplified.
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Percorso guidato
5:53Graph of Sine and Cosine Function
Constant Multiple Rule in Integration
This rule states that constants can be factored out of integrals, simplifying calculations. For example, ∫a·f(x) dx = a ∫f(x) dx. Applying this rule is essential when coefficients appear after using trigonometric identities.
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Percorso guidato
05:56Additional Rules for Indefinite Integrals
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Use any method to evaluate the integrals in Exercises 55–66.
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