`int_0^1sqrt((1-x)/(1+x))dx`

Options

A. `(pi)/(2)`
B. `((pi)/(2)-1)`
C. `((pi)/(2)+1)`
D. none of these

Explanation

`I=int_(0)^(1){(sqrt(1-x))/(sqrt(1+x))xx(sqrt(1-x))/(sqrt(1-x))}dx=int_(0)^(1)((1-x))/(sqrt(1-x^(2)))dx`.
`=int_(0)^(1)(dx)/(sqrt(1-x^(2)))+(1)/(2)int_(0)^(1)(-2x)/(sqrt(1-x^(2)))dx=[sin^(-1)x]_(0)^(1)+(1)/(2)int_(1)^(0)(1)/(sqrtt))dt`, where `(1-x^(2))=t`.
`=[sin^(-1)1-sin^(-1)0]-(1)/(2)*int_(0)^(1)(1)/(sqrt(t))dt=((pi)/(2)-0)-(1)/(2)*[(t^(1//2))/((t//2))]_(0)^(1)=((pi)/(2)-1)`.