निम्न फलनों का समाकलन कीजिए - `x cos^(-1)x`

Explanation

माना `I=intx cos^(-1)x dx`
माना `cos^(-1)x=t`
`rArr" "x=cost" "rArr" "dx=-sint dt`
`therefore" "I=int x cos^(-1)x dx =-intt cos t. sin tdt`
`=-(1)/(2)intt.2 sin t cost dt=-(1)/(2)intt.sin 2t dt`
`I=-(1)/(2[t int sin 2t dt- int{(d)/(dt)(t).int sin 2t dt}dt]`
`=-(1)/(2)[t((-cos2t))/(2)+int1.(cos2t)/(2)dt]`
`=(1)/(4)t cos 2t - (1)/(4)int cos 2tdt`
`=(1)/(4)t cos 2t-(1)/(4)(sin2t)/(2)+C`
`=(1)/(4)tcos 2t-(1)/(8)sin2t+C`
`=(1)/(4)t (2cos^(2)t-1)-(1)/(8).2sin t cost+C`
`=(1)/(4)t(2cos^(2)t-1)-(1)/(4)(1-cos^(2)t)^(1//2)cost+C`
`therefore" "int x cos^(-1)x dx=(1)/(4)cos^(-1)x(2x^(2)-1)`
`" "-(1)/(4)(1-x^(2))^(1//2)x+C`
`=(1)/(4)(2x^(2)-1)cos^(-1)x-(1)/(4)xsqrt(10x^(2))+C`