9–61. Trigonometric integrals Evaluate the following integrals.
60. ∫ from 0 to π/8 of √(1 - cos8x) dx

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Recognize that the integral involves a square root of a trigonometric expression, specifically √(1 - cos(8x)). To simplify this, use the trigonometric identity 1 - cos(θ) = 2sin²(θ/2). Rewrite the integrand as √(2sin²(4x)).

Simplify √(2sin²(4x)) to √2 * |sin(4x)|. Since the limits of integration are from 0 to π/8 and sin(4x) is non-negative in this interval, |sin(4x)| simplifies to sin(4x). The integrand becomes √2 * sin(4x).

Factor out the constant √2 from the integral. The integral now becomes √2 ∫ from 0 to π/8 of sin(4x) dx.

To integrate sin(4x), use the standard formula for the integral of sin(kx), which is -1/k * cos(kx). Here, k = 4, so the integral of sin(4x) is -1/4 * cos(4x).

Apply the limits of integration (from 0 to π/8) to the antiderivative -1/4 * cos(4x). Multiply the result by √2 to complete the evaluation of the definite integral.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Key identities, such as the Pythagorean identity and double angle formulas, can simplify integrals involving trigonometric functions. For example, the identity 1 - cos(θ) = 2sin²(θ/2) can be useful in transforming the integrand in the given integral.

Integration Techniques

Integration techniques are methods used to evaluate integrals that may not be solvable by basic antiderivatives. Techniques such as substitution, integration by parts, and trigonometric substitution can be applied to simplify the integral. In this case, a substitution involving the sine function may help in evaluating the integral of √(1 - cos(8x)).

Definite Integrals

Definite integrals represent the accumulation of quantities, calculated over a specific interval. They are denoted with limits of integration, such as from 0 to π/8 in this problem. The Fundamental Theorem of Calculus connects differentiation and integration, allowing us to evaluate definite integrals by finding the antiderivative and applying the limits to obtain a numerical result.

9–61. Trigonometric integrals Evaluate the following integrals.60. ∫ from 0 to π/8 of √(1 - cos8x) dx