How do you find the antiderivative of # (e^(2x))(sqrt(1+3e^(2x)))#?

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Answer 1 · 2016-08-01T11:57:44

#=1/9 (1 + 3 e^(2x))^(3/2) + C#

there's a simple pattern here

#d/dx (1 + 3 e^(2x))^(3/2)#

#=3/2 (1 + 3 e^(2x))^(1/2) * 6 e^(2x)#

#=9 e^(2x) (1 + 3 e^(2x))^(1/2) #

so # d/dx( 1/9 (1 + 3 e^(2x))^(3/2) )= e^(2x) (1 + 3 e^(2x))^(1/2)#

so

#int \ e^(2x) (1 + 3 e^(2x))^(1/2) \ dx = int \ d/dx ( 1/9 (1 + 3 e^(2x))^(3/2) )\ dx#

#=1/9 (1 + 3 e^(2x))^(3/2) + C#

of course you might wish to use a sub #u = e^(2x)# etc, but this way allows you to do it pretty much in your head.