Evaluate the integrals in Exercises 23–32.
∫₀^π √(1 - cos(2x)) dx

검증된 단계별 안내

Recognize that the integral involves the expression \( \sqrt{1 - \cos(2x)} \). Recall the trigonometric identity \( 1 - \cos(2x) = 2\sin^2(x) \). Use this to rewrite the integrand.

Substitute the identity into the integral to get \( \int_0^{\pi} \sqrt{2\sin^2(x)} \, dx \). Since the square root of a square is the absolute value, rewrite the integrand as \( \int_0^{\pi} \sqrt{2} |\sin(x)| \, dx \).

Consider the behavior of \( \sin(x) \) on the interval \( [0, \pi] \). Since \( \sin(x) \) is nonnegative on this interval, \( |\sin(x)| = \sin(x) \). Simplify the integral accordingly.

Factor out the constant \( \sqrt{2} \) from the integral to get \( \sqrt{2} \int_0^{\pi} \sin(x) \, dx \).

Evaluate the integral \( \int_0^{\pi} \sin(x) \, dx \) by finding the antiderivative of \( \sin(x) \), which is \( -\cos(x) \), and then apply the Fundamental Theorem of Calculus by substituting the limits \( 0 \) and \( \pi \).

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주요 개념

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, the identity for cos(2x), such as cos(2x) = 1 - 2sin²(x), helps simplify the integrand √(1 - cos(2x)) into a more manageable form for integration.

Definite Integration

Definite integration calculates the exact area under a curve between two limits, here from 0 to π. Understanding how to evaluate integrals with specific bounds is essential to find the numerical value of the integral after simplifying the integrand.

Simplification of Radicals in Integrals

Simplifying expressions under a square root before integrating can make the integral easier to solve. Recognizing that √(1 - cos(2x)) can be rewritten using trigonometric identities reduces complexity and allows the use of standard integral formulas.

Evaluate the integrals in Exercises 23–32.∫₀^π √(1 - cos(2x)) dx