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If x in[-1/2,1] then sin^(-1)(sqrt(3)/(2...

If `x in[-1/2,1] then sin^(-1)(sqrt(3)/(2)x-1/2sqrt(1-x^(2)))`

A

`sin^(-1)1/2 -sin^(-1)x`

B

`sin^(-1)x-(pi)/(6)`

C

`sin^(-1)x+(pi)/(6)`

D

none of these

Text Solution

Verified by Experts

Let `x=sin theta` then
`-1/2lexle1rarr-1/2lesin theta le 1 rarr -(pi)/(6)le theta le (pi)/(2)`
Now
`sin^(-1)sqrt(3)/(2)x-1/2sqrt(1-x^(2))`
`=sin^(-1){sin(theta-(pi)/(6))}`
`=theta-(pi)/(6)`
`=sin^(-1)x-(pi)/(6)`
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Knowledge Check

  • inte^(sin^(-1)x)((x+sqrt(1-x^2))/(sqrt(1-x^2)))dx=

    A
    `xe^(sin^(-1)x)+c`
    B
    `-xe^(sin^(-1)x)+c`
    C
    `e^(tan^(-1)x)+c`
    D
    `-e^(sin^(-1)x)+c`

  • If int(2x-sqrt(sin^(-1)x))/(sqrt(1-x^(2)))dx=C-2sqrt(1-x^(2))-(2)/(3)sqrt(f(x)) then f(x) is equal to

    A
    `sin^(-1)x`
    B
    `2sin^(-1)x`
    C
    `(sin^(-1)x)^(3)`
    D
    `3(sin^(-1)x)^(3)`

  • The complete solution set of the equation sin^(-1) sqrt((1+x)/(2))-sqrt(2-x)=cot^(-1)(tan sqrt(2-x))-sin^(-1) sqrt((1-x)/(2)) is :

    A
    `[2-(pi^(2))/(4),1]`
    B
    `[1-(pi^(2))/(4), 1]`
    C
    `[2-(pi^(2))/(4), 0]`
    D
    `[-1, 1]`

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