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inttan^(-1)sqrt((1-x)/(1+x))dx=...

`inttan^(-1)sqrt((1-x)/(1+x))dx=`

A

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

B

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

C

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

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
Put x=cos t and dx =-sin t dt. Then,
`I=inttan^(-1)sqrt((1-cost)/(1+cost))*(-sint)dt=inttan^(-1)sqrt((2sin^(2)((t)/(2)))/(2cos^(2)((t)/(2))))(-sint)dt`
`=-inttan^(-1)("tan"(t)/(2))(-sint)dt=(1)/(2)intunderset(I)(t)underset(II)((sint))dt`.
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