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

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

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integral of (ln(x))/(xsqrt(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

= \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

\frac{( x + 1 )}{( x - 1 )}

f (x) = x^{2} + 3, (- 1, 4)

\int_{2}^{t} \frac{1}{5 x ^{2}} d x

\frac{d}{d x} ((x + 2)^{2} - 3 x)

\int \frac{8 x ^{2} - 7 x + 16}{x ^{3} + 4 x} d x

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