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

The integral of (ln(x))/(x(sqrt(1+ln(x)))) is 2/3 (1+ln(x))^{3/2}-2sqrt(1+ln(x))+C

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

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

\frac{2}{3} (1 + ln (x))^{\frac{3}{2}} - 2 1 + ln (x) + C

Solution steps

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

Apply u-substitution

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

Expand

\frac{u - 1}{u} : u - \frac{1}{u}

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

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

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

= \frac{2}{3} u^{\frac{3}{2}} - 2 u

Substitute back

u = 1 + ln (x)

= \frac{2}{3} (1 + ln (x))^{\frac{3}{2}} - 2 1 + ln (x)

Add a constant to the solution

= \frac{2}{3} (1 + ln (x))^{\frac{3}{2}} - 2 1 + ln (x) + C

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What is the integral of (ln(x))/(x(sqrt(1+ln(x)))) ?
The integral of (ln(x))/(x(sqrt(1+ln(x)))) is 2/3 (1+ln(x))^{3/2}-2sqrt(1+ln(x))+C