Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.7.69

Capitolo 7, Problema 7.7.69

Evaluate the integrals in Exercises 31–78.
69. ∫dy/(y√(4y²-1))

Guida verificata passo dopo passo

Identify the integral to solve: \(\int \frac{dy}{y \sqrt{4y^{2} - 1}}\).

Recognize that the integrand contains a square root of a quadratic expression in the form \(\sqrt{a^{2}y^{2} - b^{2}}\), suggesting a trigonometric substitution.

Use the substitution \(y = \frac{1}{2} \sec(\theta)\), which implies \(4y^{2} - 1 = \tan^{2}(\theta)\), and compute \(dy\) in terms of \(d\theta\).

Rewrite the integral entirely in terms of \(\theta\), simplifying the expression using trigonometric identities to get an integral in terms of \(\theta\).

Integrate with respect to \(\theta\), then substitute back to express the answer in terms of the original variable \(y\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Integration of Rational Functions Involving Square Roots

This concept involves integrating functions where the integrand contains a rational expression combined with a square root of a quadratic expression. Recognizing the form helps in choosing appropriate substitution methods or trigonometric identities to simplify the integral.

Video consigliato:

07:01

Integrals Involving Natural Logs: Substitution

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving square roots of quadratic expressions by substituting variables with trigonometric functions. For example, expressions like √(a²x² - b²) can be simplified using secant or tangent substitutions to transform the integral into a more manageable form.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

Algebraic Manipulation and Simplification

Before applying substitution, it is often necessary to manipulate the integrand algebraically, such as factoring or rewriting terms, to identify the best substitution strategy. Simplifying the expression under the square root and the entire integrand can make the integration process more straightforward.

Video consigliato:

05:25

Determine Continuity Algebraically

Pratica correlata

Domanda del libro di testo

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.

120. y = x^(sin x)

views

Domanda del libro di testo

Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.

sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞

cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1

tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1

sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1

csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1

coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1

Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.

65. sech⁻¹(3/5)

views

Domanda del libro di testo

In Exercises 139–142, find the length of each curve.

141. y = ln(cos(x)) from x = 0 to x = π/4.