Domanda del libro di testo
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
63. ∫ dx/(x² - 2x - 15)
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.RE.6
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
6. ∫ (2 − sin 2θ)/cos² 2θ dθ
Guida verificata passo dopo passo1
Rewrite the integral by separating the terms in the numerator: ∫ (2 - sin(2θ)) / cos²(2θ) dθ = ∫ (2 / cos²(2θ)) dθ - ∫ (sin(2θ) / cos²(2θ)) dθ.
For the first term, recognize that 2 / cos²(2θ) can be rewritten using the secant function: 2 / cos²(2θ) = 2 sec²(2θ). The integral of sec²(x) is a standard result, so ∫ 2 sec²(2θ) dθ = (1/2) tan(2θ) + C.
For the second term, rewrite sin(2θ) / cos²(2θ) as (sin(2θ) / cos(2θ)) * (1 / cos(2θ)). Notice that (sin(2θ) / cos(2θ)) is equivalent to tan(2θ), and 1 / cos(2θ) is sec(2θ). Thus, the term becomes ∫ tan(2θ) sec(2θ) dθ.
To integrate ∫ tan(2θ) sec(2θ) dθ, use the substitution u = cos(2θ), which implies du = -2 sin(2θ) dθ. Rewrite the integral in terms of u and solve.
Combine the results of the two integrals, simplify, and include the constant of integration, C, to express the final solution.
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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 partial fraction decomposition. Understanding these methods is crucial for evaluating complex integrals, as they allow for simplification and manipulation of the integrand to make integration feasible.
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Percorso guidato
06:18
Integration by Parts for Definite Integrals
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. Key identities, such as the Pythagorean identities and angle sum formulas, can simplify integrals involving trigonometric functions. Recognizing and applying these identities is essential for transforming the integrand into a more manageable form.
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7:17Verifying Trig Equations as Identities
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 standard form that is easier to evaluate. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative.
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07:33Euler's Method
Pratica correlata
Domanda del libro di testo
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
22. ∫ tan³ 5θ dθ
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Domanda del libro di testo
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
51. ∫ (from 0 to π/4) sin⁵(4θ) dθ
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Domanda del libro di testo
118. Two worthy integrals
b. Let f be any positive continuous function on the interval [0, π/2]. Evaluate
∫ from 0 to π/2 of [f(cos x) / (f(cos x) + f(sin x))] dx.
(Hint: Use the identity cos(π/2 − x) = sin x.)
(Source: Mathematics Magazine 81, 2, Apr 2008)
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Domanda del libro di testo
120. Equal volumes
a. Let R be the region bounded by the graph of f(x) = x^(-p) and the x-axis, for x ≥ 1. Let V₁ and V₂ be the volumes of the solids generated when R is revolved about the x-axis and the y-axis, respectively, if they exist. For what values of p (if any) is V₁ = V₂?
b. Repeat part (a) on the interval [0, 1].
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Domanda del libro di testo
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
32. ∫ csc²(6x) cot(6x) dx
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