see the picture of the solution . like my answer?

I = e^2x(1/x -1/2x² _{)dx       lim 1 to 2}

 put  2x =t

differentiate wrt t

 dx = dt/2

now integral becomes

 I= e^t (2/t -2/t² ) /2 dt              lim 2 to 4

 I= e^t (1/t -1/t² )dt         lim 2 to 4         

  e^x {f(x)  + f¹(x)} dx = e^x f(x)                 ( this is identity )

 so by using this  we get 

  I=(e^t /t )  lim 2 to 4   

  I= (e⁴ -2e² )/4        ans