Domanda del libro di testo
7–84. Evaluate the following integrals.
64. ∫ (ln(ax))/x dx, where a ≠ 0
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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 8.4.23
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
23. ∫ 1/(25 - x²)^(3/2) dx
Guida verificata passo dopo passo1
Step 1: Recognize the integral involves a term of the form \( a^2 - x^2 \), which suggests using the trigonometric substitution \( x = a \sin(\theta) \). Here, \( a = 5 \) because \( 25 = 5^2 \). Substitute \( x = 5 \sin(\theta) \), and compute \( dx = 5 \cos(\theta) d\theta \).
Step 2: Substitute \( x = 5 \sin(\theta) \) and \( dx = 5 \cos(\theta) d\theta \) into the integral. The term \( 25 - x^2 \) becomes \( 25 - 25 \sin^2(\theta) \), which simplifies to \( 25 \cos^2(\theta) \) using the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
Step 3: Rewrite the integral in terms of \( \theta \): \( \int \frac{1}{(25 \cos^2(\theta))^{3/2}} \cdot 5 \cos(\theta) d\theta \). Simplify the denominator \( (25 \cos^2(\theta))^{3/2} \) to \( 125 \cos^3(\theta) \). The integral becomes \( \int \frac{5 \cos(\theta)}{125 \cos^3(\theta)} d\theta \).
Step 4: Simplify the fraction \( \frac{5 \cos(\theta)}{125 \cos^3(\theta)} \) to \( \frac{1}{25 \cos^2(\theta)} \). The integral now becomes \( \int \frac{1}{25 \cos^2(\theta)} d\theta \), which can be rewritten as \( \frac{1}{25} \int \sec^2(\theta) d\theta \).
Step 5: Evaluate \( \int \sec^2(\theta) d\theta \), which is a standard integral equal to \( \tan(\theta) \). After finding \( \tan(\theta) \), use the substitution \( \theta = \arcsin(x/5) \) to express the result back in terms of \( x \).
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Trigonometric Substitution
Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of quadratic expressions. By substituting a variable with a trigonometric function, such as x = a sin(θ) or x = a tan(θ), the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(a² - x²), √(x² + a²), or √(x² - a²).
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Percorso guidato
6:04
Introduction to Trigonometric Functions
Integral of Trigonometric Functions
Understanding the integrals of trigonometric functions is essential for evaluating integrals after substitution. Common integrals include ∫ sin(θ) dθ = -cos(θ) + C and ∫ cos(θ) dθ = sin(θ) + C. Familiarity with these integrals allows for the effective evaluation of the transformed integral, leading to the final solution after reverting back to the original variable.
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6:04Introduction to Trigonometric Functions
Pythagorean Identity
The Pythagorean identity, sin²(θ) + cos²(θ) = 1, is a fundamental relationship in trigonometry that is often used in conjunction with trigonometric substitution. This identity helps simplify expressions involving trigonometric functions, especially when converting back to the original variable after integration. Recognizing how to manipulate this identity is crucial for solving integrals that arise from trigonometric substitutions.
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7:17Verifying Trig Equations as Identities
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Domanda del libro di testo
9–61. Trigonometric integrals Evaluate the following integrals.
38. ∫ tan⁵θ sec⁴θ dθ
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Domanda del libro di testo
79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.
82. ∫ (sin⁻¹(ax)) / x² dx, a > 0
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Domanda del libro di testo
94. The family f(x) = 1/xᵖ revisited Consider the family of functions f(x) = 1/xᵖ, where p is a real number.
For what values of p does the integral ∫(1 to ∞) 1/xᵖ dx exist?
What is its value when it exists?
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Domanda del libro di testo
5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
12. (2x² + 3)/((x² - 8x + 16)(x² + 3x + 4))
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Domanda del libro di testo
71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.
71. ∫[x/(ax + b)] dx (Hint: u = ax + b.)
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