Evaluate: (i) `int(4x+2)sqrt(x^2+x+1)\ dx` (ii) `int(4x+3)/(sqrt(2x^2+3x+1))\ dx`

Explanation

`(i) I=int(4x+2)sqrt(x^2+x+1)\ dx`
taking `x^2+x+1=t`
`=>(2x+1)dx=dt`
`=>(4x+2)dx=2dt`
so, `I=int sqrt t 2 dt`
`=2 int sqrt t dt`
`= 2* 2/3 t^(3/2) +c`
`=4/3 (x^2+x+1)^(3/2) +c`

`(ii) I=int(4x+3)/(sqrt(2x^2+3x+1))\ dx`
taking `sqrt(2x^2+3x+1)=t`
`=>1/(2sqrt(2x^2+3x+1))(4x+3) dx=dt`
so,` I=int 2dt`
`=2t+c`
`=2sqrt(2x^2+3x+1)`