`sqrt(1+3x-x^(2))`

Explanation

माना `I=intsqrt(1+2x-x^(2))dx`
`=intsqrt(-(x^(2)-3x-1))dx`
`=intsqrt(-[x^(2)-3x+((3)/(2))^(2)-((3)/(3))^(2)-1])dx`
`=intsqrt(-[(x-(3)/(2))^(2)-(9)/(4)-1])dx`
`=intsqrt(-[(x-(3)/(2))^(2)-(9+4)/(4)])dx`
`=intsqrt([-(x-(3)/(2))^(2)-(13)/(3)])dx`
`=intsqrt(((sqrt(13))/(2))^(2)-(x-(3)/(2))^(2))dx`
`=(x-(3)/(2))/(2)sqrt(1+3x-x^(2))+(13)/(4xx2)sin^(-1)((x-(3)/(2))/((sqrt(13))/(2)))`
`rArr" "I=(2x-3)/(4)sqrt(1+3x-x^(2))+(13)/(8)sin^(-1)((2x-3)/(sqrt(13)))+C`