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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.RE.48
Capitolo 8, Problema 8.RE.48

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
48. ∫ sin(3x) cos⁶(3x) dx

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Step 1: Recognize that the integral involves a product of trigonometric functions, specifically sin(3x) and cos⁶(3x). To simplify, consider using a substitution method. Let u = cos(3x), which implies that du = -3sin(3x) dx.
Step 2: Rewrite the integral in terms of u. Substitute sin(3x) dx with -du/3 and cos⁶(3x) with u⁶. The integral becomes: ∫ sin(3x) cos⁶(3x) dx = -1/3 ∫ u⁶ du.
Step 3: Apply the power rule for integration to evaluate ∫ u⁶ du. Recall that the power rule states: ∫ uⁿ du = uⁿ⁺¹ / (n+1) + C, where n ≠ -1.
Step 4: Substitute back u = cos(3x) into the result obtained from the integration step to express the solution in terms of x.
Step 5: Add the constant of integration (C) to complete the solution, as this is an indefinite integral.

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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 identities. Understanding these methods is essential for evaluating complex integrals, as they allow for simplification and manipulation of the integrand to facilitate integration.
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Integration by Parts for Definite Integrals

Trigonometric Identities

Trigonometric identities are equations that relate the angles and sides of triangles, and they are crucial in simplifying integrals involving trigonometric functions. For example, the product-to-sum identities can transform products of sine and cosine into sums, making integration more manageable. Familiarity with these identities is vital for solving integrals like ∫ sin(3x) cos⁶(3x) dx.
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Substitution Method

The substitution method is a technique used in integration where a new variable is introduced to simplify the integral. By substituting a part of the integrand with a single variable, the integral can often be transformed into a more straightforward form. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative.
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Euler's Method
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2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

29. ∫ cos⁴ x/sin⁶ x dx

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66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

66. Let f(x) = cos(x²).

a. Find a Midpoint Rule approximation to ∫[-1 to 1] cos(x²) dx using n = 30 subintervals.

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66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

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2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

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32. ∫ csc²(6x) cot(6x) dx

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