Evaluate: (i) `int1/(sqrt(tan^(-1)x)(1+x^2)\ dx` (ii) `int(sqrt(tanx))/(sinxcosx)\ dx`

Explanation

`(i) I=int1/(sqrt(tan^(-1)x)(1+x^2)\ dx`
taking `sqrt(tan^(-1)x)=t`
`=1/sqrt(tan^(-1)x)1/(1+x^2) dx`
`=2dt`
putting these values,integration becomes,
`I=int 2dt`
`=2t+c`
`=2sqrt(tan^(-1)x)+c`

`(ii) I=int(sqrt(tanx))/(sinxcosx)\ dx`
`=int sqrt(tanx)/(cos^2x)/(sinx*cosx)/cos^2x dx`
`=int(sqrt tanx sec^2x)/tanx dx`
taking `tanx=t`
`=>sec^2xdx=dt`
so. `I=int sqrtt/t dt`
`=2sqrtt+c`
`=2 sqrttanx +c`