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

1 Answer
Jun 26, 2016

Answer:

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

Explanation:

#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#