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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.3.15
Capitolo 8, Problema 8.3.15

9–61. Trigonometric integrals Evaluate the following integrals.
15. ∫ sin³x cos²x dx

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Step 1: Recognize that the integral involves powers of sine and cosine. To simplify, use trigonometric identities. Specifically, split the odd power of sine: sin³x = sin²x * sinx, and recall the Pythagorean identity sin²x = 1 - cos²x.
Step 2: Rewrite the integral as ∫ (1 - cos²x) * sinx * cos²x dx. This allows us to express the integral in terms of cosx, making substitution easier.
Step 3: Perform a substitution. Let u = cosx, which implies that du = -sinx dx. Substitute these into the integral, replacing cosx with u and sinx dx with -du.
Step 4: After substitution, the integral becomes ∫ (1 - u²) * u² * (-du). Simplify this expression to ∫ (-u² + u⁴) du.
Step 5: Integrate term by term. Use the power rule for integration: ∫ uⁿ du = uⁿ⁺¹ / (n+1). After integrating, back-substitute u = cosx to express the result in terms of x.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. They are essential for simplifying integrals involving trigonometric functions. For example, the Pythagorean identity sin²x + cos²x = 1 can be used to rewrite integrals in a more manageable form.
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Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric substitution. For the integral ∫ sin³x cos²x dx, using substitution or recognizing patterns in the integrand can simplify the process of finding the antiderivative.
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Power Reduction Formulas

Power reduction formulas are used to express powers of sine and cosine in terms of first-degree functions. These formulas, such as sin²x = (1 - cos(2x))/2, help in simplifying integrals involving higher powers of trigonometric functions. Applying these formulas can make the integration of functions like sin³x cos²x more straightforward.
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9–61. Trigonometric integrals Evaluate the following integrals.

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102–106. Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s) = ∫[0 to ∞] e^(-st) f(t) dt, where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^(-t), the following improper integral is evaluated using integration by parts:

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