Urgent) Integrate x cos^-1 x

Let I = \int x \cos^{- 1} x d x = \cos^{- 1} x \times \int x d x - \int [\frac{d}{d x} (\cos^{- 1} x) \times \int x d x] d x + C = \frac{x^{2}}{2} \cos^{- 1} x - \int [- \frac{1}{\sqrt{1 - x^{2}}} \times \frac{x^{2}}{2}] d x + C = \frac{x^{2}}{2} \cos^{- 1} x + \frac{1}{2} \int \frac{x^{2}}{\sqrt{1 - x^{2}}} d x + C Let I_{1} = \int \frac{x^{2}}{\sqrt{1 - x^{2}}} d x put x = \sin θ \Rightarrow d x = \cos θ d θ So, I_{1} = \int \frac{\sin^{2} θ}{\sqrt{1 - \sin^{2} θ}} \cos θ d θ = \int \sin^{2} θ d θ = \frac{1}{2} \int (1 - \cos 2 θ) d θ = \frac{1}{2} [θ - \frac{\sin 2 θ}{2}] = \frac{1}{2} [θ - \sin θ \cos θ] = \frac{1}{2} [\sin^{- 1} x - x \sqrt{1 - x^{2}}] Hence, I = \frac{x^{2}}{2} \cos^{- 1} x + \frac{1}{4} [\sin^{- 1} x - x \sqrt{1 - x^{2}}] + C

Urgent) Integrate x cos-1 x