Evaluate: (i) `int(1+sinx)/(sqrt(x-cosx))\ dx` (ii) `int1/(sqrt(1-x^2)(sin^(-1)x)^2)\ dx`

Explanation

`(i)`Given integral,
`I=int(1+sinx)/(sqrt(x-cosx))\ dx`
Let
`t=x−cosx`
`=>dt=(1+sinx)dx`
so,
`I=int(1+sinx)/(sqrt(x-cosx))\ dx`
`=int(dt)/sqrt t `
`=int t^(-1/2)`
`=t^(1/2)/(1/2)+c`
`=2 sqrt t +c`
`=2 sqrt(x-cos x)+c`
where `t=x-cosx`


`(ii) `Given integral is
`I=int1/(sqrt(1-x^2)(sin^(-1)x)^2)\ dx`
taking `sin^-1x=t`
`=>1/sqrt(1-x^2) dx=dt`
putting these values we get,
`I=int 1/t^2 dt`
`=-1/t +c`
`=-1/sin^-1x+c`