What is the integral of 1/(2xln(3x)) ?

The integral of 1/(2xln(3x)) is 1/2 ln|ln(3x)|+C

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integral of 1/(2xln(3x))

\int \frac{1}{2 x ln ( 3 x )} d x

\frac{1}{2} ln ∣ ln (3 x) ∣ + C

Solution steps

\int \frac{1}{2 x ln ( 3 x )} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{x ln ( 3 x )} d x

Apply u-substitution

= \frac{1}{2} \cdot \int \frac{1}{u ln ( u )} d u

Apply u-substitution

= \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} ln ∣ v ∣

Substitute back

= \frac{1}{2} ln ∣ ln (3 x) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ ln (3 x) ∣ + C

What is the integral of 1/(2xln(3x)) ?
The integral of 1/(2xln(3x)) is 1/2 ln|ln(3x)|+C