What is the integral of 1/(x(9ln(x)+8)) ?

The integral of 1/(x(9ln(x)+8)) is 1/9 ln|9ln(x)+8|+C

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integral of 1/(x(9ln(x)+8))

\int \frac{1}{x ( 9 ln ( x ) + 8 )} d x

\frac{1}{9} ln ∣ 9 ln (x) + 8 ∣ + C

Solution steps

\int \frac{1}{x ( 9 ln ( x ) + 8 )} d x

Apply u-substitution

: \int \frac{1}{9 u} d u

= \int \frac{1}{9 u} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{9} ln ∣ u ∣

Substitute back

u = 9 ln (x) + 8

= \frac{1}{9} ln ∣ 9 ln (x) + 8 ∣

Add a constant to the solution

= \frac{1}{9} ln ∣ 9 ln (x) + 8 ∣ + C

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What is the integral of 1/(x(9ln(x)+8)) ?
The integral of 1/(x(9ln(x)+8)) is 1/9 ln|9ln(x)+8|+C