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

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

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integral of 5/(xln(x))

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

5 ln ∣ ln (x) ∣ + C

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 ln ∣ u ∣

Substitute back

u = ln (x)

= 5 ln ∣ ln (x) ∣

Add a constant to the solution

= 5 ln ∣ ln (x) ∣ + C

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