What is the integral of 9/(xln(5x)) ?

The integral of 9/(xln(5x)) is 9ln|ln(5x)|+C

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integral of 9/(xln(5x))

\int \frac{9}{x ln ( 5 x )} d x

9 ln ∣ ln (5 x) ∣ + C

Solution steps

\int \frac{9}{x ln ( 5 x )} d x

Take the constant out:

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

= 9 \cdot \int \frac{1}{x ln ( 5 x )} d x

Apply u-substitution

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

= 9 \cdot \int \frac{1}{u ln ( u )} d u

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 9 ln ∣ v ∣

Substitute back

= 9 ln ∣ ln (5 x) ∣

Add a constant to the solution

= 9 ln ∣ ln (5 x) ∣ + C

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What is the integral of 9/(xln(5x)) ?
The integral of 9/(xln(5x)) is 9ln|ln(5x)|+C