Integrate:-

{(1/(logx))-(1/(logx)²)} for upper limit e² and lower limit 0.

I = {1/logx -1/(logx)² }dx

multiply numerator & denominator by x

I = x{1/logx-1/(logx)²}dx /x

put logx = t

    dx/x = dt      & x=e^t

now

        I = e^t{1/t - 1/t²}dt       

        I = e^t{ f(t) + f¹(t) }

we have a direct formula for e^x{f(x) + f¹(x)}dx = e^xf(x)

 so I = e^t f(t) 

        =e^t/t = (x / logx)       lim 0 to e²

        =0 - e²/2