Evaluate: (i) `intx^x(1+logx)\ dx` (ii) `intx^(2x)\ (1+logx)\ dx`

Explanation

`(i) I=intx^x(1+logx)\ dx`
taking `x^x=t`
`=>e^(xlogx) =t`
`=> e^(xlogx) (x xx 1/x +logx)dx=dt`
`=>x^x(1+logx)dx=dt`
putting these, our integration becomes,
`I=int dt`
`=t+c`
`=x^x+c`

`(ii) I=intx^(2x)\ (1+logx)\ dx`
`=int x^x x^x |1+logx|dx`
taking `x^x=t`
`=>e^(xlogx) =t`
`=> e^(xlogx) (x xx 1/x +logx)dx=dt`
`=>x^x(1+logx)dx=dt`
putting these, our integration becomes,
`I=int tdt`
`=t^2/2+c`
`=x^(2x)/2+c`