Evaluate the integral from [(2/square rt 2),0] of dx/sqaure root of (4-x^2)
To evaluate the integral ∫(2/√2, 0) dx / √(4 - x^2), we can use trigonometric substitution.
1. First, let's make a substitution: x = 2sinθ.
This choice is based on the identity sin^2θ + cos^2θ = 1, which resembles our expression 4 - x^2.
2. Find the differential of x in terms of θ: dx = 2cosθ dθ.
3. Substitute these values in the integral:
∫(2/√2, 0) (2cosθ dθ) / √(4 - (2sinθ)^2).
4. Simplify the expression inside the square root:
√(4 - (2sinθ)^2) = √(4 - 4sin^2θ) = √(4cos^2θ) = 2|cosθ|.
5. Since we have |cosθ| in the denominator, we need to consider the absolute value of cosθ.
6. Now, rewrite the integral:
∫(2/√2, 0) (2cosθ dθ) / (2|cosθ|).
7. Cancel out the common factors and simplify the integral:
∫(2/√2, 0) dθ.
8. Integrate with respect to θ:
∫dθ = θ.
9. Evaluate the integral from the given limits:
θ ∣(2/√2, 0) = 2/√2 - 0 = √2.
Therefore, the value of the integral ∫(2/√2, 0) dx / √(4 - x^2) is √2.