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int(dx)/(sqrt(2+2x-x^(2)))=?...

`int(dx)/(sqrt(2+2x-x^(2)))=?`

A

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

B

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

C

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

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`(2+2x-x^(2))=3-(1+x^(2)-2x)=(sqrt(3))^(2)-(x-x)^(2)`
`therefore I=int(dx)/(sqrt((sqrt(3))^(2)-(x-1)^(2)))=int(dt)/(sqrt((sqrt(3))^(2)-t^(2))), "where"(x-1)=t and dx=dt`
`=sin^(-1)(t)/(sqrt(3))+C =sin^(-1)((x-1))/(sqrt(3))+C`
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