Evaluate: `int(xcos^(-1)x)/(sqrt(1-x^2))\ dx`

Explanation

` " Let I " = int (x cos ^(-1)x)/(sqrt(1-x^(2)))dx`
` rArr I = int cos^(-1) x. (x)/(sqrt(1-x^(2)))dx`
`I= cos^(-1) x int (x)/(sqrt(1-x^(2)))dx`
`- int [(d)/(dx) (cos^(-1)x) int (x)/(sqrt(1-x^(2)))dx ]dx`
`" Let " 1-x^(2) =t^(2) rArr x dx =- dt`
`:. I = cos^(-1) x int (1)/(t) ((-t)/(1))dt`
` - int [(d)/(dx) (cos^(-1) x) int (1)/(t) ((-t)/(1)) dt]dx`
` =cos^(-1) x(-t) -int [((-1))/(sqrt(1-x^(2)))(-t) ]dx`
`=-sqrt(1-x^(2)).cos^(-1) x- int ((sqrt(1-x^(2)))/(sqrt(1-x^(2))))dx`
` =-sqrt(1-x^(2)).cos^(-1) x-int 1 dx`
` =- sqrt(1-x^(2)).cos^(-1)x-x +C`