What is the integral of-5/(xln(x^3)) ?

The integral of-5/(xln(x^3)) is -5/3 ln(3|ln(x)|)+C

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integral of-5/(xln(x^3))

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

- \frac{5}{3} ln (3 ∣ ln (x) ∣) + C

Solution steps

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

Apply u-substitution

= - 5 \cdot \int \frac{1}{3 u ln ( u )} d u

Take the constant out:

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

= - 5 \cdot \frac{1}{3} \cdot \int \frac{1}{u ln ( u )} d u

Apply u-substitution

= - 5 \cdot \frac{1}{3} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - 5 \cdot \frac{1}{3} ln ∣ v ∣

Substitute back

= - 5 \cdot \frac{1}{3} ln ln (x^{3})

Simplify

- 5 \cdot \frac{1}{3} ln ln (x^{3}) : - \frac{5}{3} ln (3 ∣ ln (x) ∣)

= - \frac{5}{3} ln (3 ∣ ln (x) ∣)

Add a constant to the solution

= - \frac{5}{3} ln (3 ∣ ln (x) ∣) + C

What is the integral of-5/(xln(x^3)) ?
The integral of-5/(xln(x^3)) is -5/3 ln(3|ln(x)|)+C