What is the integral of 1/(xln(x^5)) ?

The integral of 1/(xln(x^5)) is 1/5 ln(5|ln(x)|)+C

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integral of 1/(xln(x^5))

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

\frac{1}{5} ln (5 ∣ ln (x) ∣) + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{5} ln ∣ v ∣

Substitute back

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

Simplify

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

= \frac{1}{5} ln (5 ∣ ln (x) ∣)

Add a constant to the solution

= \frac{1}{5} ln (5 ∣ ln (x) ∣) + C

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What is the integral of 1/(xln(x^5)) ?
The integral of 1/(xln(x^5)) is 1/5 ln(5|ln(x)|)+C