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

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

Grade:12

1 Answers

vikas askiitian expert
509 Points
12 years ago

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 =et dt lim 1 to 4

now expression becomes

 I= t1/2 .et 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


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