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

Verified step by step guidance

1. 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.
2. 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 \( \int_0^{\pi} \sqrt{2} |\sin(x)| \, dx \).
3. 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.
4. Factor out the constant \( \sqrt{2} \) from the integral to get \( \sqrt{2} \int_0^{\pi} \sin(x) \, dx \).
5. 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 \( \pi \).

Key Concepts

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.