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int(dx)/(sqrt(1+2x-3x^(2)))...

`int(dx)/(sqrt(1+2x-3x^(2)))`

A

`(1)/(sqrt(3))sin^(-1)((3x-1)/(2))+C`

B

`(1)/(sqrt(2))sin^(-1)((2x-1)/(2))+C`

C

`(1)/(sqrt(3))sin^(-1)((2x+1)/(2))+C`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`(1+2x-3x^(2))=-3(x^(2)-(2)/(3)x-(1)/(3))=3{(4)/(9)-(x^(2)-(2)/(3)x+(1)/(9))}`
`=3{((2)/(3))^(2)-(x-(1)/(3))^(2)}.`
`therefore I=(1)/(sqrt(3)).int (dx)/(sqrt(((2)/(3))^(2)-(x-(1)/(3))^(2)))=(1)/(sqrt(3)).int(dt)/(sqrt(((2)/(3))^(2)-t^(2)))=(1)/(sqrt(3))sin^(-1)""((t)/((2)/(3)))+C`
`=(1)/(sqrt(3))sin^(-1)""((3t)/(2))+C=(1)/(sqrt(3))sin^(-1)((3x-1))/(2)+C.`
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