`(xcos^(-1)x)/(sqrt(1-x^(2)))`

Explanation

माना `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`
माना `1-x^(2)=t^(2)" "rArr" "xdx=-tdt`
`therefore" "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(10x^(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-int1 dx`
`=-sqrt(1-x^(2)).cos^(-1)x-x+C`