Definite integral 1 to 2 e^x² dx = a then find definite integral e to e^4 (lnx)^1/2 dx

we have to find integral (lnx)^1/2 lim e to e⁴ ....let it be I...

I= (lnx)^1/2dx     lim e to e⁴ 

now put lnx =t   

after differentiating we get 

  dx =e^t dt lim 1 to 4

now expression becomes

 I= t^1/2 .e^t dt             lim 1 to 4

now put t = u² , dt =2udu

 now I= 2u² e^u² du     lim 1 to 2

now replacing variable u with x

 I= 2x² e^x² dx      lim 1to 2      ............1

a=e^x² dx  lim 1 to 2           (given)

 integrating RHS and taking 1 as first function

a =xe^x² - 2x² e^x² dx    lim 1 to 2

putting value of I in above

 we get I= xe^x² -a    lim 1 to 2

      I= 2e⁴ -e -a            ans