Mathematics: Find the integral, using techniques from this or the previous chapter. ∫ln 4x+5| dx.

None of them pass thorugh the point (-1,5) however if you meant (1,-5) then y=3x-8 would be the answer. Step-by-step explanation:...

Find the integral, using techniques from this - id4545967, josephmartinelli5

ruth4941

15.01.2022

$\int ln|4x+5| dx=(x+\frac{5}{4} )(ln|4x+5|)+c\\$

Step-by-step explanation:

The formula to integrate a natural logarithm is

$\int lnu du =uln|u|-u+c$

in this case

u=4x+5

and the derivative:

du=4dx

thus

$dx=\frac{du}{4}$

we have are asked for the integral:

$\int ln|4x+5| dx$

so replacing u=4x+5 and $dx=\frac{du}{4}$

$\int ln|u|\frac{du}{4}$

which is the same as

$\frac{1}{4} \int ln|u|du$

using the formula we have:

$\frac{1}{4}\int ln|u|du =\frac{1}{4}(uln|u|-u)+c$

and since u=4x+5

$=\frac{1}{4} [(4x+5)ln|4x+5|]+c\\=(x+\frac{5}{4} )(ln|4x+5|)+c\\$

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