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int(dx)/(sqrt(x-x^2)) is equal to...

`int(dx)/(sqrt(x-x^2))` is equal to

A

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

B

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

C

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

D

None of these

Text Solution

Verified by Experts

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