Question

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

Answer 1

`\int ln|4x+5| dx=(x+(5)/(4) )(ln|4x+5|)+c\\`

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=(du)/(4)`

we have are asked for the integral:

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

so replacing
`u=4x+5` and
`dx=(du)/(4)`

`\int ln|u|(du)/(4)`

which is the same as

`(1)/(4) \int ln|u|du`

using the formula we have:

`(1)/(4)\int ln|u|du =(1)/(4)(uln|u|-u)+c`

and since
`u=4x+5`

`=(1)/(4) [(4x+5)ln|4x+5|]+c\\=(x+(5)/(4) )(ln|4x+5|)+c\\`