How do you integrate #(e^(sqrt(1+3x)))dx#?

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Answer 1 · 2016-06-26T10:27:35

#= 2/3 *e^sqrt(1+3x) ( sqrt(1+3x) - 1) + C#

#int \ (e^(sqrt(1+3x))) \ dx#

sub #p = sqrt(1+3x), \qquad dp = 3/(2 sqrt(1+3x)) dx = 3/(2p) dx#

#\implies 2/3 int \ p e^p \ dp#

IBP

#u = p, u' = 1#
#v' = e^p, v = e^p#

#\implies 2/3 ( p e^p - int \ e^p \ dp)#

#= 2/3 ( p e^p - e^p ) + C#

#= 2/3 *e^p ( p - 1) + C#

#= 2/3 *e^sqrt(1+3x) ( sqrt(1+3x) - 1) + C#