What is the integral of (ln(x))/(x(ln(x)+1)) ?

The integral of (ln(x))/(x(ln(x)+1)) is ln(x)+1-ln|ln(x)+1|+C

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integral of (ln(x))/(x(ln(x)+1))

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

ln (x) + 1 - ln ∣ ln (x) + 1 ∣ + C

Solution steps

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

Apply u-substitution

= \int \frac{u}{u + 1} d u

Apply u-substitution

= \int \frac{v - 1}{v} d v

Expand

\frac{v - 1}{v} : 1 - \frac{1}{v}

= \int 1 - \frac{1}{v} d v

Apply the Sum Rule:

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

= \int 1 d v - \int \frac{1}{v} d v

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= v - ln ∣ v ∣

Substitute back

= ln (x) + 1 - ln ∣ ln (x) + 1 ∣

Add a constant to the solution

= ln (x) + 1 - ln ∣ ln (x) + 1 ∣ + C

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