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

The integral of 1/(xln(sqrt(x))) is 2ln|ln(x)|+C

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integral of 1/(xln(sqrt(x)))

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

2 ln ∣ ln (x) ∣ + C

Solution steps

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

Simplify

\frac{1}{x ln ( x )} : \frac{2}{x ln ( x )}

= \int \frac{2}{x ln ( x )} d x

Take the constant out:

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

= 2 \cdot \int \frac{1}{x ln ( x )} d x

Apply u-substitution

= 2 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = ln (x)

= 2 ln ∣ ln (x) ∣

Add a constant to the solution

= 2 ln ∣ ln (x) ∣ + C

What is the integral of 1/(xln(sqrt(x))) ?
The integral of 1/(xln(sqrt(x))) is 2ln|ln(x)|+C