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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 8.7.82
Capitolo 8, Problema 8.7.82

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

Guida verificata passo dopo passo
1
Identify the integral to solve: \(\int \frac{\sin^{-1}(ax)}{x^{2}} \, dx\), where \(a > 0\).
Consider using integration by parts, since the integrand is a product of functions involving an inverse trigonometric function and a power of \(x\). Let \(u = \sin^{-1}(ax)\) and \(dv = \frac{1}{x^{2}} dx\).
Compute the derivatives and integrals needed for integration by parts: find \(du = \frac{a}{\sqrt{1 - a^{2}x^{2}}} dx\) and \(v = -\frac{1}{x}\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), substituting the expressions for \(u\), \(v\), and \(du\).
Simplify the resulting integral, which will likely require a second table look-up or substitution to evaluate \(\int \frac{1}{x \sqrt{1 - a^{2}x^{2}}} dx\). Use a table of integrals or a computer algebra system to find this integral.

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Inverse Trigonometric Functions

Inverse trigonometric functions, like sin⁻¹(x), are the inverses of the standard trigonometric functions and return an angle whose sine is x. Understanding their properties and derivatives is essential for integrating expressions involving these functions.
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Derivatives of Other Inverse Trigonometric Functions

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation. It is useful when integrating products of functions, such as an inverse trigonometric function multiplied by an algebraic expression, by reducing the integral into simpler parts.
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Integration by Parts for Definite Integrals

Use of Integral Tables and Computer Algebra Systems

Integral tables provide standard forms of integrals that can simplify complex integration problems. Computer algebra systems (CAS) help verify solutions and handle complicated integrals, ensuring accuracy and saving time.
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Integration Using Partial Fractions
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Domanda del libro di testo

87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

88. Evaluate ∫ dx/(2 + cos x).

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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

7–84. Evaluate the following integrals.

62. ∫ from 0 to π/2 √(1 + cosθ) dθ

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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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Domanda del libro di testo

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

23. ∫ 1/(25 - x²)^(3/2) dx

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