Answer:
[tex]\int ln|4x+5| dx=(x+\frac{5}{4} )(ln|4x+5|)+c\\[/tex]
Step-by-step explanation:
The formula to integrate a natural logarithm is
[tex]\int lnu du =uln|u|-u+c[/tex]
in this case
[tex]u=4x+5[/tex]
and the derivative:
[tex]du=4dx[/tex]
thus
[tex]dx=\frac{du}{4}[/tex]
we have are asked for the integral:
[tex]\int ln|4x+5| dx [/tex]
so replacing [tex]u=4x+5[/tex] and [tex]dx=\frac{du}{4}[/tex]
[tex]\int ln|u|\frac{du}{4}[/tex]
which is the same as
[tex]\frac{1}{4} \int ln|u|du[/tex]
using the formula we have:
[tex]\frac{1}{4}\int ln|u|du =\frac{1}{4}(uln|u|-u)+c[/tex]
and since [tex]u=4x+5[/tex]
[tex]=\frac{1}{4} [(4x+5)ln|4x+5|]+c\\=(x+\frac{5}{4} )(ln|4x+5|)+c\\[/tex]
