Evaluate: `int_0^1(2-x^2)/((1+x)sqrt(1-x^2))dx`

Answer

`(pi)/2`

Explanation

Put `x=sintheta`. So `dx=cos theta d theta`
`:.` Given integral
`=int_(0)^(pi//2)((2-sin^(2)theta)cos theta d theta)/((1+sin theta) cos theta)`
`=int_(0)^(pi//2)(1-sin theta+1/(1+sin theta))d theta`
`=|theta + cos theta|_(0)^(pi//2)+int_(0)^(pi//2)(d theta)/(1+sin theta)`
`=(pi)/2-1+int_(0)^(pi//2)(1-sin theta)/(cos^(2)theta) d theta`
`(pi)/2-1+int_(0)^(pi//2) (sec^(2) theta -sec theta tan )d theta`
`=(pi)/2-1+|tan theta -sec theta|_(0)^(pi//2)`
`=(pi)/2-1+lim_(theta to pi//2)(sin theta-1)/(cos theta)-(sin 0-1)/(cos 0)`
`=(pi)/2-1+lim_(theta to pi//2) (cos theta)/(sin theta) +1`
`=(pi)/2`