7–84. Evaluate the following integrals.
62. ∫ from 0 to π/2 √(1 + cosθ) dθ

Verified step by step guidance

1. Step 1: Recognize that the integral involves a square root and a trigonometric function. To simplify √(1 + cosθ), use the trigonometric identity 1 + cosθ = 2cos²(θ/2). This substitution will simplify the square root.
2. Step 2: Substitute 1 + cosθ with 2cos²(θ/2). The integral becomes ∫ from 0 to π/2 √(2cos²(θ/2)) dθ, which simplifies further to ∫ from 0 to π/2 √2 * |cos(θ/2)| dθ.
3. Step 3: Since θ ranges from 0 to π/2, cos(θ/2) is non-negative, so |cos(θ/2)| = cos(θ/2). The integral now becomes ∫ from 0 to π/2 √2 * cos(θ/2) dθ.
4. Step 4: Perform a substitution to simplify the integral further. Let u = θ/2, which implies du = (1/2)dθ. Adjust the limits of integration: when θ = 0, u = 0; when θ = π/2, u = π/4. The integral becomes ∫ from 0 to π/4 √2 * cos(u) * 2 du.
5. Step 5: Simplify the integral to ∫ from 0 to π/4 2√2 * cos(u) du. Now, integrate cos(u) with respect to u, which results in sin(u). Substitute the limits of integration to evaluate the result.

Key Concepts

Definite Integral

A definite integral represents the signed area under a curve between two specified limits. In this case, the integral from 0 to π/2 indicates that we are calculating the area under the curve of the function √(1 + cosθ) from θ = 0 to θ = π/2. Understanding the properties of definite integrals, such as the Fundamental Theorem of Calculus, is essential for evaluating them.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. In this integral, the expression √(1 + cosθ) can be simplified using the half-angle identity, which states that 1 + cosθ = 2cos²(θ/2). Recognizing and applying these identities can simplify the integration process significantly.

Substitution Method

The substitution method is a technique used in integration to simplify the integrand by changing variables. For the integral ∫ √(1 + cosθ) dθ, substituting θ/2 can transform the integral into a more manageable form. This method is particularly useful when dealing with complex expressions, allowing for easier evaluation of the integral.