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

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

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integral of (ln(ln(ln(x))))/(xln(x))

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

ln (ln (x)) ln (ln (ln (x))) - ln (ln (x)) + C

Solution steps

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

Apply u-substitution

= \int \frac{ln ( ln ( u ) )}{u} d u

Apply u-substitution

= \int ln (v) d v

Apply Integration By Parts

= v ln (v) - \int 1 d v

\int 1 d v = v

= v ln (v) - v

Substitute back

= ln (ln (x)) ln (ln (ln (x))) - ln (ln (x))

Add a constant to the solution

= ln (ln (x)) ln (ln (ln (x))) - ln (ln (x)) + C

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