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

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

A

`2tan^(-1)(x+sqrt(1-x^(2)))+c`

B

`tan^(-1)(x+sqrt(1-x^(2)))+c`

C

`2tan^(-1)(x-sqrt(1-x^(2)))+c`

D

`2cot^(-1)(x+sqrt(1-x^(2)))+c`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `x=sin theta, dx = cos theta d theta`
`therefore" "i=int((cos theta-sin theta)cos theta d theta)/(cos theta(1+sin thetacostheta))`
`=2int((cos theta-sin theta)d theta)/(1+(cos theta+sin theta))`
`=2tan^(-1)(cos theta+sin theta)+c`
`=2 tan^(-1)(x+sqrt(1-x^(2)))+c`
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