Evaluate ∫ (1 - x^2)^(1/2) dx. What is the result?

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The integral ∫ (1 - x^2)^(1/2) dx requires a method of integration that often involves trigonometric substitution due to its square root structure.

To evaluate this integral, we can use the substitution x = sin(θ), which then transforms the expression inside the square root:

1 - x^2 = 1 - sin^2(θ) = cos^2(θ), leading to (1 - x^2)^(1/2) = cos(θ). Furthermore, dx becomes cos(θ) dθ. Thus, the integral can be rewritten as:

∫ cos(θ) cos(θ) dθ = ∫ cos^2(θ) dθ.

Using the identity cos^2(θ) = (1 + cos(2θ))/2, the integral becomes:

∫ (1 + cos(2θ))/2 dθ = (1/2) ∫ dθ + (1/2) ∫ cos(2θ) dθ.

The first integral yields (1/2)θ, and the second integral yields (1/4)sin(2θ) + C when evaluated. Combining these results gives: