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Evaluate int(dx)/(sqrt(x(1-2x)))....

Evaluate `int(dx)/(sqrt(x(1-2x)))`.

A

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

B

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

C

`(1)/(sqrt(2))sin^(-1)(4x+1)+C`

D

`(1)/(sqrt(2))sin^(-1)(4x-1)+C`

Text Solution

Verified by Experts

The correct Answer is:
D
`x(1-2x)=(x-2x^(2))=2((x)/(2)-x^(2))=2[(1)/(16)-(x^(2)-(x)/(2)+(1)/(16))]=2[((1)/(4))^(2)-(x-(1)/(4))^(2)]`
`therefore I=(1)/(sqrt(2))int(dx)/(sqrt(((1)/(4))^(2)-(x-(1)/(4))^(2)))=(1)/(sqrt(2)).int(dx)/(sqrt(((1)/(4))^(2)-t^(2))),"where" (x-(1)/(4))=t`
`=(1)/(sqrt(2))sin^(-1)""(t)/(((1)/(4)))+C=(1)/(sqrt(2))sin^(-1)4t+C=(1)/(sqrt(2))sin^(-1){4(x-(1)/(4))}+C`
`=(1)/(sqrt(2))sin^(-1)(4x-1)+C.`
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