9–61. Trigonometric integrals Evaluate the following integrals.
59. ∫ from 0 to π/2 of √(1 - cos2x) dx

Verified step by step guidance

1. Recognize that the integral involves a trigonometric expression under a square root. Use the trigonometric identity: $1\; -\; ext\{cos\}(2x)\; =\; 2\; ext\{sin\}^2(x)$ to simplify the expression inside the square root.
2. Substitute the identity into the integral: $\int \_\{0\}^\{\pi /2\}\; ext\{\surd \}(1\; -\; ext\{cos\}(2x))\; dx\; =\; \int \_\{0\}^\{\pi /2\}\; ext\{\surd \}(2\; ext\{sin\}^2(x))\; dx$. Factor out the constant $\surd 2$ from the square root.
3. Simplify further: $\int \_\{0\}^\{\pi /2\}\; ext\{\surd \}(2\; ext\{sin\}^2(x))\; dx\; =\; \surd 2\; \int \_\{0\}^\{\pi /2\}\; |\; ext\{sin\}(x)|\; dx$. Since $x$ is in the interval $[0,\; \pi /2]$, $|\; ext\{sin\}(x)|\; =\; ext\{sin\}(x)$.
4. The integral now becomes: $\surd 2\; \int \_\{0\}^\{\pi /2\}\; ext\{sin\}(x)\; dx$. Use the standard formula for the integral of $ext\{sin\}(x)$, which is $-\; ext\{cos\}(x)$.
5. Evaluate the definite integral: $\surd 2\; [-\; ext\{cos\}(x)]\_\{0\}^\{\pi /2\}$. Substitute the limits of integration into the expression to complete the solution.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. In this problem, the identity sin²(x) + cos²(x) = 1 can be used to simplify the integrand √(1 - cos²(2x)), which is equivalent to sin(2x). Recognizing and applying these identities is crucial for evaluating trigonometric integrals.

Integration Techniques

Integration techniques refer to various methods used to compute integrals, such as substitution, integration by parts, and trigonometric substitution. In this case, recognizing that the integral can be simplified using the identity mentioned allows for easier integration. Mastery of these techniques is essential for solving complex integrals effectively.

Definite Integrals

Definite integrals represent the area under a curve between two specified limits. In this problem, the integral is evaluated from 0 to π/2, which means we need to find the total area under the curve of the function √(1 - cos²(2x)) within this interval. Understanding how to compute definite integrals is fundamental in calculus, as it connects to real-world applications such as calculating areas and accumulated quantities.