The value of the integral ∫ d x 1 + x x - x 2 is equal to (where, C is the constant of integration)

Step-by-step solution

Let, I = ∫ d x x 1 + x 1 - x Put, x = s i n 2 θ and d x = 2 sin ⁡ θ cos ⁡ θ d θ ⇒ I = ∫ 2 sin ⁡ θ cos ⁡ θ d θ sin ⁡ θ 1 + sin ⁡ θ cos ⁡ θ = 2 ∫ d θ 1 + sin ⁡ θ = 2 ∫ 1 - sin ⁡ θ c o s 2 θ d θ = 2 tan ⁡ θ - sec ⁡ θ + C = 2 x 1 - x - 1 1 - x + C = 2 x - 1 1 - x + C