How do you find the integral #int_0^13dx/(root3((1+2x)^2)# ?

1 Answer
Sep 7, 2014

By substitution,
#int_0^{13}1/{root3{(1+2x)^2}}dx=3#

Let #u=1+2x#.
By taking the derivative,
#{du}/{dx}=2#
By taking the reciprocal,
#{dx}/{du}=1/2#
By multiply by #du#,
#dx={du}/2#

Since #x# goes from 0 to 13, #u# goes from 1 to 27.
Now, we can rewrite the integral in terms of #u#.
#int_0^{13}1/{root3{(1+2x)}}dx =int_1^{27}1/{root3{u^2}}{du}/2#
by simplifying,
#=1/2int_1^{27}u^{-2/3}du#
by Power Rule,
#=1/2[3u^{1/3}]_1^{27} =3/2[root3{27}-root3{1}]=3/2cdot 2=3#