What is the integral of (ln(ln(5x)))/x ?

The integral of (ln(ln(5x)))/x is ln(5x)ln(ln(5x))-ln(5x)+C

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integral of (ln(ln(5x)))/x

\int \frac{ln ( ln ( 5 x ) )}{x} d x

ln (5 x) ln (ln (5 x)) - ln (5 x) + C

Solution steps

\int \frac{ln ( ln ( 5 x ) )}{x} d x

Apply u-substitution

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

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

Apply u-substitution

: \int ln (v) d v

= \int ln (v) d v

Apply Integration By Parts

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

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

\int 1 d v = v

= v ln (v) - v

Substitute back

= ln (5 x) ln (ln (5 x)) - ln (5 x)

Add a constant to the solution

= ln (5 x) ln (ln (5 x)) - ln (5 x) + C

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What is the integral of (ln(ln(5x)))/x ?
The integral of (ln(ln(5x)))/x is ln(5x)ln(ln(5x))-ln(5x)+C