What is the integral of 6/(xln(4x)) ?

The integral of 6/(xln(4x)) is 6ln|ln(4x)|+C

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integral of 6/(xln(4x))

\int \frac{6}{x ln ( 4 x )} d x

6 ln ∣ ln (4 x) ∣ + C

Solution steps

\int \frac{6}{x ln ( 4 x )} d x

Take the constant out:

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

= 6 \cdot \int \frac{1}{x ln ( 4 x )} d x

Apply u-substitution

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

= 6 \cdot \int \frac{1}{u ln ( u )} d u

Apply u-substitution

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

= 6 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 6 ln ∣ v ∣

Substitute back

= 6 ln ∣ ln (4 x) ∣

Add a constant to the solution

= 6 ln ∣ ln (4 x) ∣ + C

\int 3 x^{2} (x^{3} - 5)^{2} d x

\int \frac{5 x}{x ^{4} + 1} d x

\int x^{- \frac{3}{5}} d x

\int x + ∣ x ∣ d x

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

What is the integral of 6/(xln(4x)) ?
The integral of 6/(xln(4x)) is 6ln|ln(4x)|+C