What is the integral of 5/(xln(6x)) ?

The integral of 5/(xln(6x)) is 5ln|ln(6x)|+C

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integral of 5/(xln(6x))

\int \frac{5}{x ln ( 6 x )} d x

5 ln ∣ ln (6 x) ∣ + C

Solution steps

\int \frac{5}{x ln ( 6 x )} d x

Take the constant out:

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

= 5 \cdot \int \frac{1}{x ln ( 6 x )} d x

Apply u-substitution

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

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

Apply u-substitution

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

= 5 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 5 ln ∣ v ∣

Substitute back

= 5 ln ∣ ln (6 x) ∣

Add a constant to the solution

= 5 ln ∣ ln (6 x) ∣ + C

\int_{0}^{3} x 16 - x^{2} d x

\int \frac{1}{x ^{2} x ^{2} - 3} d x

x \to 0 lim (- x ln (x) - x)

\int x \cdot x d x

\frac{1}{x - 1}

What is the integral of 5/(xln(6x)) ?
The integral of 5/(xln(6x)) is 5ln|ln(6x)|+C