What is the integral of 1/(x(1+ln(x))(2+ln(x))) ?

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

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integral of 1/(x(1+ln(x))(2+ln(x)))

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

ln ∣ 1 + ln (x) ∣ - ln ∣ ln (x) + 2 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u ( u + 1 )} d u

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

Take the partial fraction of

\frac{1}{u ( u + 1 )} : \frac{1}{u} - \frac{1}{u + 1}

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

= ln ∣ u ∣ - ln ∣ u + 1 ∣

Substitute back

u = 1 + ln (x)

= ln ∣ 1 + ln (x) ∣ - ln ∣ 1 + ln (x) + 1 ∣

Simplify

= ln ∣ 1 + ln (x) ∣ - ln ∣ ln (x) + 2 ∣

Add a constant to the solution

= ln ∣ 1 + ln (x) ∣ - ln ∣ ln (x) + 2 ∣ + C

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