What is the integral of (ln(x)*ln(ln(x)))/x ?

The integral of (ln(x)*ln(ln(x)))/x is 1/2 ln^2(x)ln(ln(x))-(ln^2(x))/4+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (ln(x)*ln(ln(x)))/x

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

\frac{1}{2} ln^{2} (x) ln (ln (x)) - \frac{ln ^{2} ( x )}{4} + C

Solution steps

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

Apply u-substitution

: \int u ln (u) d u

= \int u ln (u) d u

Apply Integration By Parts

: \frac{1}{2} u^{2} ln (u) - \int \frac{u}{2} d u

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

\int \frac{u}{2} d u = \frac{u ^{2}}{4}

= \frac{1}{2} u^{2} ln (u) - \frac{u ^{2}}{4}

Substitute back

u = ln (x)

= \frac{1}{2} ln^{2} (x) ln (ln (x)) - \frac{ln ^{2} ( x )}{4}

Add a constant to the solution

= \frac{1}{2} ln^{2} (x) ln (ln (x)) - \frac{ln ^{2} ( x )}{4} + C

\int \frac{sin ( ln ( 3 x ) )}{x} d x

\int \frac{1}{( 3 - 2 x )} d x

\int (1 - x) (1 - x^{2}) d x

\int (x - 1)^{2} (x - 2) d x

\int \frac{x ^{3} + 1}{x x} d x

What is the integral of (ln(x)*ln(ln(x)))/x ?
The integral of (ln(x)*ln(ln(x)))/x is 1/2 ln^2(x)ln(ln(x))-(ln^2(x))/4+C