Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.7.69

Chapter 7, Problem 7.7.69

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

Verified step by step guidance

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 \(y\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

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Related Practice

Textbook Question

13. When is a polynomial f(x) of at most the order of a polynomial g(x) as x→∞? Give reasons for your answer.

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Textbook Question

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)

Textbook Question

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

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

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