Evaluate the integrals in Exercises 53–76.
63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))

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

Recognize that the integrand involves a square root of a quadratic expression, \(\sqrt{4y^{2} - 1}\), which suggests a trigonometric substitution to simplify the square root.

Use the substitution \(y = \frac{1}{2} \sec(\theta)\), because \(4y^{2} - 1 = 4 \left( \frac{1}{2} \sec(\theta) \right)^{2} - 1 = \sec^{2}(\theta) - 1 = \tan^{2}(\theta)\), which simplifies the square root to \(\tan(\theta)\).

Compute \(dy\) in terms of \(d\theta\): \(dy = \frac{1}{2} \sec(\theta) \tan(\theta) d\theta\), and rewrite the integral entirely in terms of \(\theta\).

Change the limits of integration from \(y\) to \(\theta\) by solving \(y = \frac{1}{2} \sec(\theta)\) for the given bounds, then integrate the resulting expression with respect to \(\theta\).

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주요 개념

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Definite Integrals and Limits of Integration

A definite integral calculates the net area under a curve between two specific points, called limits of integration. Understanding how to apply these limits correctly is essential for evaluating the integral's numerical value over the given interval.

Integration Techniques for Rational Functions Involving Roots

Integrals involving expressions like 1/(y√(4y² - 1)) often require substitution methods or recognizing standard integral forms. Techniques such as trigonometric substitution or algebraic manipulation simplify the integrand to a more manageable form.

Domain and Validity of the Integrand

Before integrating, it is crucial to analyze the domain where the integrand is defined, especially when involving square roots and denominators. Ensuring the expression under the root is non-negative and the denominator is non-zero avoids invalid or undefined values.

Evaluate the integrals in Exercises 53–76.63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))