What is the integral of (ln(ln(2x)))/(2x) ?

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

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integral of (ln(ln(2x)))/(2x)

\int \frac{ln ( ln ( 2 x ) )}{2 x} d x

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

Solution steps

\int \frac{ln ( ln ( 2 x ) )}{2 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{ln ( ln ( 2 x ) )}{x} d x

Apply u-substitution

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

Apply u-substitution

= \frac{1}{2} \cdot \int ln (v) d v

Apply Integration By Parts

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

\int 1 d v = v

= \frac{1}{2} (v ln (v) - v)

Substitute back

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

Add a constant to the solution

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

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