7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
12. ∫[1/2 to 1] √(1 - x²)/x² dx

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Recognize that the integral involves a square root of the form √(1 - x²), which suggests using a trigonometric substitution. Specifically, let x = sin(θ), so that dx = cos(θ)dθ and √(1 - x²) becomes √(1 - sin²(θ)) = cos(θ).

Substitute x = sin(θ) into the integral. The limits of integration will also change: when x = 1/2, θ = arcsin(1/2), and when x = 1, θ = arcsin(1). Rewrite the integral in terms of θ.

After substitution, the integral becomes ∫[arcsin(1/2) to arcsin(1)] (cos²(θ)/sin²(θ)) dθ. Simplify the integrand using trigonometric identities, such as cos²(θ) = 1 - sin²(θ) or cot²(θ) = cos²(θ)/sin²(θ).

Evaluate the simplified integral. This may involve breaking it into simpler parts or using additional trigonometric identities to integrate terms like cot²(θ).

After finding the antiderivative, substitute back the original limits of integration (arcsin(1/2) and arcsin(1)) to compute the definite integral. Simplify the result as needed.

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주요 개념

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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 = sin(θ) or x = tan(θ), the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(1 - x²) or √(x² - 1).

Definite Integrals

A definite integral calculates the accumulation of a quantity, represented as the area under a curve, between two specified limits. In this case, the integral ∫[1/2 to 1] indicates that we are interested in the area from x = 1/2 to x = 1. Evaluating definite integrals often involves finding the antiderivative of the function and then applying the Fundamental Theorem of Calculus to compute the difference between the values at the upper and lower limits.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry that states sin²(θ) + cos²(θ) = 1. This identity is crucial when performing trigonometric substitutions, as it allows for the simplification of expressions involving square roots. For example, when substituting x = sin(θ), the identity helps to express √(1 - x²) in terms of cos(θ), facilitating the integration process.

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.12. ∫[1/2 to 1] √(1 - x²)/x² dx