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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
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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