Evaluate the integrals in Exercises 23–32.
∫_{π/2}^{3π/4} √(1 - sin(2x)) dx

검증된 단계별 안내

Start by examining the integrand: \(\sqrt{1 - \sin(2x)}\). Recognize that the expression inside the square root can be simplified using trigonometric identities.

Recall the double-angle identity for sine: \(\sin(2x) = 2\sin x \cos x\). However, a more useful identity here is the one involving cosine of a double angle: \(1 - \sin(2x) = 1 - 2\sin x \cos x\).

Use the identity \(1 - \sin(2x) = (\cos x - \sin x)^2\) because \((\cos x - \sin x)^2 = \cos^2 x - 2\sin x \cos x + \sin^2 x = 1 - 2\sin x \cos x = 1 - \sin(2x)\).

Rewrite the integral as \(\int_{\pi/2}^{3\pi/4} \sqrt{(\cos x - \sin x)^2} \, dx = \int_{\pi/2}^{3\pi/4} |\cos x - \sin x| \, dx\). Consider the absolute value carefully over the interval to determine the sign of \(\cos x - \sin x\).

Split the integral if necessary based on where \(\cos x - \sin x\) changes sign, then integrate \(\cos x - \sin x\) or its negative accordingly. Use the antiderivatives \(\int \cos x \, dx = \sin x\) and \(\int \sin x \, dx = -\cos x\) to evaluate the integral.

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주요 개념

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Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It involves evaluating the integral function at the upper and lower bounds and subtracting these values. Understanding the properties of definite integrals is essential for solving integrals with given limits.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Identities like sin(2x) = 2sin(x)cos(x) help simplify complex expressions inside integrals, making them easier to evaluate.

Integration Techniques for Trigonometric Functions

Integrating functions involving trigonometric expressions often requires substitution, simplification using identities, or recognizing standard integral forms. Mastery of these techniques allows one to transform complicated integrals into solvable forms.

Evaluate the integrals in Exercises 23–32.∫_{π/2}^{3π/4} √(1 - sin(2x)) dx