Evaluate the following integral using trigonometric substitution. \( \int_{0}^{1} \frac{d x}{\sqrt{4-x^{2}}} \) What substitution will be the most helpful for evaluating this integral? A. \( x=2 \sec \theta \) B. \( x=2 \tan \theta \) C. \( x=2 \sin \theta \)

To evaluate the integral   ∫₀¹ dx/√(4 − x²) using a trigonometric substitution, we look for a substitution that will simplify the square root in the denominator. Since the integrand contains √(4 − x²), a common substitution is to let   x = 2 sinθ. This substitution is chosen because it transforms the square root as follows: 1. Express x in terms of θ: x = 2 sinθ. 2. Differentiate: dx = 2 cosθ dθ. 3. Substitute into the square root:   √(4 − x²) = √(4 − 4 sin²θ) = √(4(1 − sin²θ)) = √(4 cos²θ) = 2|cosθ|. 4. For the range of x from 0 to 1, θ will fall in [0, π/6]. In this interval, cosθ is positive, so |cosθ| = cosθ. Thus, the substitution simplifies the integrand:   dx/√(4 − x²) = (2 cosθ dθ)/(2 cosθ) = dθ. The limits of integration also change: - When x = 0, 0 = 2 sinθ ⇒ θ = 0. - When x = 1, 1 = 2 sinθ ⇒ sinθ = 1/2 ⇒ θ = π/6. The integral becomes   ∫₀^(π/6) dθ = θ|₀^(π/6) = π/6. Thus, the substitution that is most helpful is:   x = 2 sinθ. So, the correct choice is C.