What is the integral of (ln(2x))/(xln(x)) ?

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

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integral of (ln(2x))/(xln(x))

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

ln (2) ln ∣ ln (x) ∣ + ln (x) + C

Solution steps

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

Apply u-substitution

= \int \frac{ln ( 2 ) + u}{u} d u

Expand

\frac{ln ( 2 ) + u}{u} : \frac{ln ( 2 )}{u} + 1

= \int \frac{ln ( 2 )}{u} + 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{ln ( 2 )}{u} d u + \int 1 d u

\int \frac{ln ( 2 )}{u} d u = ln (2) ln ∣ u ∣

\int 1 d u = u

= ln (2) ln ∣ u ∣ + u

Substitute back

u = ln (x)

= ln (2) ln ∣ ln (x) ∣ + ln (x)

Add a constant to the solution

= ln (2) ln ∣ ln (x) ∣ + ln (x) + C

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