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Evaluate the following definite integral...

Evaluate the following definite integrals `:`
`int_(0)^(1) sqrt((1-x)/(1+x)) dx `

A

`(pi)/(2)+1`

B

`(pi)/(2)-1`

C

`-1`

D

1

Text Solution

Verified by Experts

The correct Answer is:
B
`I=int_(0)^(1)sqrt((1-x)/(1+x))dx=int_(0)^(1)(1-x)/(sqrt(1+x^(2)))`
`=int_(0)^(1)(1)/(sqrt(1-x^(2)))dx-int_(0)^(1)(x)/(sqrt(1-x^(2)))dx`
`=[sin^(-1)x]_(0)^(1)+int_(0)^(1)(t)/(t)dt`
where , `t^(2)=1-x^(2)rArrtdt =-x dx]`
`=(sin^(-1)-sin^(-1)0)+[t]_(1)^(0)=pi//2-1`
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