If the value of the integral Ι = ∫ 0 1 d x x + 1 - x 2 is equal to π k , then the value of k is equal to

Step-by-step solution

Let, x = sin ⁡ θ ,   ∀ θ ∈ 0 , π 2 ⇒ d x = cos ⁡ θ d θ So, Ι = ∫ 0 π / 2 cos ⁡ θ d θ sin ⁡ θ + cos ⁡ θ Applying a + b - x property and adding, we get 2 Ι = ∫ 0 π / 2 cos ⁡ θ + sin ⁡ θ sin ⁡ θ + cos ⁡ θ d θ = ∫ 0 π / 2 1 d θ = θ 0 π 2 ⇒ 2 Ι = π 2 ⇒ Ι = π 4 ∴ k = 4