`intsqrt((1+x)/(1-x))dx=?`

Options

A. `sin^(-1)x+sqrt(1-x^(2))+C`
B. `sin^(-1)x+(1+x^(2))+C`
C. `sin^(-1)x-sqrt(1-x^(2))+C`
D. none of these

Answer

C

Explanation

`I=int{(sqrt(1+x))/(sqrt(1-x))xxsqrt(1+x)/(sqrt(1+x))}dx=int((1+x))/(sqrt(1-x^(2)))dx`
`=int(1)/(sqrt(1-x^(2)))dx-(1)/(2)(2x)/(sqrt(1-x^(2)))dx=sin^(-1)x-(1)/(2)int(1)/(sqrt(t))dt,where (1-x^(2))=t`
`=sin^(-1)x-sqrt(t)+C=sin^(-1)x-sqrt(1-x^(2))+C`.