What is the integral of 8/((x+3)ln(x+3)) ?

The integral of 8/((x+3)ln(x+3)) is 8ln|ln(x+3)|+C

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integral of 8/((x+3)ln(x+3))

\int \frac{8}{( x + 3 ) ln ( x + 3 )} d x

8 ln ∣ ln (x + 3) ∣ + C

Solution steps

\int \frac{8}{( x + 3 ) ln ( x + 3 )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 8 \cdot \int \frac{1}{( x + 3 ) ln ( x + 3 )} d x

Apply u-substitution

: \int \frac{1}{u ln ( u )} d u

= 8 \cdot \int \frac{1}{u ln ( u )} d u

Apply u-substitution

: \int \frac{1}{v} d v

= 8 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 8 ln ∣ v ∣

Substitute back

= 8 ln ∣ ln (x + 3) ∣

Add a constant to the solution

= 8 ln ∣ ln (x + 3) ∣ + C

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What is the integral of 8/((x+3)ln(x+3)) ?
The integral of 8/((x+3)ln(x+3)) is 8ln|ln(x+3)|+C