Evaluate `int(dx)/((1+x^(2))sqrt(1-x^(2))).`

Explanation

` I=int(dx)/((1-x^(2))sqrt(1-x^(2)))`
Put `x="sin"theta`
`:. I=int("cos"theta d theta)/((1+sin^(2)theta)(cos theta))`
`=int(d theta)/(1+sin^(2)theta)`
`=int("cosec"^(2)theta d theta)/(2+cot^(2)theta)`
Put `cot theta=t`
`:. -"cosec"^(2)theta d theta=dt`
`:. I=int(-dt)/(2+t^(2))`
`=-(1)/(sqrt(2))"tan"^(-1)(t)/(sqrt(2))+c`
`= -(1)/(sqrt(2))"tan"^(-1)(cot theta)/(sqrt(2))+c`
`=-(1)/(sqrt(2))"tan"^(-1)(sqrt(1-x^(2)))/(sqrt(2x))+c`