Question

Evaluate the definite integral exactly using the Fundamental Theorem. Simplify your answer.

∫₀¹/² 1/√(1−y²) ​dy

Answer 1

To evaluate the integral ∫₀¹/² 1/√(1−y²) ​dy, use the substitution y = sin(x) to simplify the integral to ∫₀^(π/6) dx. Evaluate the integral to get π/6 as the answer.

Step-by-step explanation:

To evaluate the definite integral ∫₀¹/² 1/√(1−y²) ​dy using the Fundamental Theorem, we can simplify the integral by using the substitution y = sin(x). This way, we have y' = cos(x) dx, and dx = dy / cos(x). Substituting these values, the integral becomes ∫₀^(π/6) 1/√(1−sin²(x)) ​(dy / cos(x)).

Using the identity sin²(x) + cos²(x) = 1, we can replace the square root term with cos(x). The integral then becomes ∫₀^(π/6) 1/cos(x) ​dy. Since dy = cos(x) dx, the integral becomes ∫₀^(π/6) dx.

Evaluating the integral, we have [x] from 0 to π/6, which simplifies to π/6 - 0 = π/6.